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PHYSIOS

CHARLES RIBORG MANN

THE UNIVERSITY OP CHICAGO

AND

GEORGE RANSOM TWISS

THE CENTRAL HIGH SCHOOL, CLEVELAND

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ROBERT O. LAW COMPANY.
PRINTERS AND BINDERS. Ch(ICAOO

MARSH. AITKEN & CURTIS COMPANY, CHICAOO

PREFACE

Within the last twenty years the methods of teaching physics '
have been revolutionized. The reaction against the loose and
desultory methods previously in vogue was started by the em-
phasis given to laboratory work in the new books which then
appeared. The movement gained impetus from the influence
brought to bear on 'the schools by the Harvard Entrance Require-
ments and the Report of the Committee of Ten. This pressure
on the schools resulted in a demand for closer observation by the
students, involving careful measurements.

As far as science is concerned, the most important result of
this introduction of laboratory work into the schools has been the
development in the public mind of a widespread recognition of
the fundamental principle that knowledge is real and living to the
individual, only when it is founded on personally observed facts
and personal experience.

Nevertheless, during the past few years it has become clear
that the present methods of applying this principle to instruction
in physics have failed to arouse in the students that real enthusi-
asm for the pursuit of the subject, which is indispensable for the
mastery of its principles. This has been found true in spite of
the fact that, in their own way, boys and giris have by nature and
disposition the keenest interest in physical phenomena.

In recognition of this fact, many attempts have been made
to develop a better system of instruction. Such a system ought
to give the mental training that has been so much emphasized;
but it ought also to inspire in the boys and giris a living enthusiasm
for the subject, and to develop in them the scientific habit of
mind, the ability to utilize their knowledge, and a just apprecia-
tion of the significance of natural phenomena. If physics can
be so taught as to develop in the student these elements of
power, so vital to his future career, there can be no doubt that
in due time its educative value will • be properiy appreciated and

its popularity restored.

iii

IV PHYSICS

This book is the contribution of two of the fraternity of teachers
toward the attainment of this end. The method of instruction
herein set forth has been developed after long experience and
much experimenting with high school and college classes.
The greater part of the book has been in manuscript for more
than two years, and has been used in connection with class
teaching. The results obtained have been so encouraging that
the book is submitted to other teachers, in the hope that it may
be of service to them.

In a recent work, President G. Stanley Hall comments at
length on the decline of interest in physics in the high schools,
and consequently also in the colleges, and he' suggests several
remedies.^ He attributes this failure of physics to "the violence
done to the nature and needs of the youthful soul by the present
methods and matters." He points out that this violence consists
in: 1. Neglect of the hero-ology of the science, of historical and
biographical references, so that the learner is not made to "feel
vividly a sense of growth." 2. "The rage to apply mathematics
to the boy's brain processes," instead of appealing to his interest
in concrete things. 3. The failure to realize that "very much
thoroughness and perfection violates the laws of youthful nature
and of groA\i;h." The young student "wants only answers that
are vague, brief, but above all suggestive." 4. Neglect of the
practical side of the subject, which is the side that appeals most
strongly to the youth. **He is chiefly interested in the 'go' of
things."

The methods of instruction which have proved helpful to us,
and which are embodied in this book, are in harmony with many of
President HalFs suggestions. We have endeavored to strengthen
the presentation of the subject, and aid the teacher in three ways:
^ I. By arousing interest. II. By developing the scientific habit of
! thought. HI. By presenting some of the principles from the
historical standpoint. Some of the ideas that have guided us in
this endeavor are the following:

I. Interest. Interest is rarely stimulated in youth by elegant

* Adolescence, by G. Stanley Hall, Vol. II, pp. 154 seq., New York,
AppletOQ, 1905.

PREFACE V

and abstract mathematical treatment; nor is it often aroused
by rigorous logical demonstrations. It is aroused by beginning
with some concrete thing that goes — ^something which is already
familiar. Interest may be sustained by basing the discussion on
these familiar and concrete things. Nothing helps more than
to have the student feel that you are discussing with him some-
thing concerning which he already knows a little, and of which he ;
has long been desirous of knowing more. The authors believe that
the adoption of an informal style and the use of arguments that i
are physical, rather than mathematical, will also be helpful; for
they have been mindful of the success of the great teachers, Fara-
day and Tyndall, in imparting scientific ideas to untrained minds
in this way. Mathematics is an excellent servant but a very
bad master; so equations are used only where they are clearly a
help to the student, and the development of each is carefully
presented with the aid of physical, rather than mathematical
concepts.

The aim has been to show the student that knowledge of physics :
enables him to answer many of the questions over which he has )
puzzled long in vain. He is approached with the attitude: What
do the forces of Nature do for us, and how do they do it? His
self -activity is stimulated by this questioning attitude of text-book
and teacher, and he is urged to investigate independently at home.
His interest is not killed at the start by attempting to cram him
with definitions of things to which he has no corresponding con-
cept, such as indestructibility, impenetrability, and the like.
Nor is he deceived by attempted definitions of undefinable con-
cepts, such as mass, force, time, and space. On the contrary,
the attempt is made to implant the concept and create the demand
for its name, or definition, which is withheld Until the need for it '
is apparent.

II. The Scientific Method. Although interest may be ob-
tained through the technical applicati^s of physics, the teaching
must not consist in descriptions of these only, any more than of
descriptions of laboratory apparatus only. The attainment of
scientific principles is always the purpose or end of the argument;
not inventions, nor yet laboratory experiments. Science must

VI PHYSICS

be shown to consist in that body of organized knowledge which
makes invention possible. Beginning arguments with inven-
tions, or general observations of phenomena, may not be the
logical order, but it is more nearly the order in which Nature
herself teaches, and the result of the argument does not lose in
definiteness, clearness, or accuracy^ provided the laboratory is
continually held up as the final court of appeal where all doubtful
questions are settled.

Each chapter in this book is a continuous argument toward
some principle or principles, and the entire book is an argument
toward the conclusions stated in the last chapter. This treatment
; is intended to develop and foster the habit of scientific thinking.
The attempt is made (1), to interest the student in observing care-
fully and accurately first the familiar things about him, and then
the things in the laboratory; (2), to interest him in detecting analo-
gies and similarities among the things observed; (3), to train him in
I keeping his mind free from bias and in drawing conclusions
( tentatively; (4), to make him see the value of verifying the con-
i elusions and accepting the result, whether it confirms or denies. his
inferences. The arguments in the various parts of the book are
not all alike; there are many forms in which the scientific method
may be used.

We have tried deliberately to give the student the impression
that science leads to no absolute results — that, at best, it is merely
a question of close approximation; of doing the best we can, and
accepting the result tentatively, until we can do better. This
attitude places the teacher also in the position of a learner and
prohibits him from making use of didactic or dogmatic statements;
for these are the bane of science as well as of other things. Science
instruction, that does not develop mental integrity, freedom of
the personal judgment, and tolerance, fails in a very vital spot.

III. History. References are given to books in which the
biographies of the great men of science may be read, and the
student is urged to read them and report. The arguments used
by some of the great thinkers have been briefly sketched, and
the methods devised by them for reaching conclusions have been
given. The attempt has been made to present them as they live

PREFACE VU

in the ideas which they have handed down to us; to picture their
mental processes and attitude, and to show how one thing leads
to another as the subject develops in the discoverer's mind.

We wish to call the attention of our colleagues to several prac-
tical points. In the first place, although each chapter is a con-
tinuous argument, the paragraphs are headed in black type,
so that the important steps are well marked; and a summary and |^
set of questions are added at the end of each chapter, to assist T
the student in fixing the subject-matter in mind. The teacher
will, we think, find these latter very helpful to his pupils in both
advance and review work.

In the second place, the continuity of the treatment is not
interrupted by the insertion of descriptions of laboratory and
lecture experiments in fine type. Judged from our own experience,
such experiments, thus inserted, confuse rather than assist the
student. It goes without saying, that we expect both laboratory (
and lecture experiments to be given in connection with tliis book;y
but every laboratory experiment made by the student, and* every
experimental demonstration by the teacher should have a definite \
relation in time, place, and subject matter to the general argument \
as presented in the text. An experiment is simply an incum-
brance and a source of distraction to the student unless its rela- {
tion to the general scheme of the lessons in the classroom is per- '
fectly obvious. A detailed description of a lecture experiment
which he has not seen is of relatively small value to the student,
and ordinarily there is no interest or profit to him in obtruding
on his attention the distracting details of setting up and manipu-
lating the apparatus. If such description of an experiment
occurs in the text book, while the teacher chooses to make it
with some other style of apparatus, different in its details, his con- /
fusion is all the worse, for his attention is distracted from the(
principle to be illustrated, and lost in the details of the apparatus.

On the other hand, when the student is to make an experi-
ment himself in the laboratory, he must be given many details in
order that he may manipulate, observe, and record successfully
and without loss of time. It is the province of the laboratory \
manual to give these details, for they can not be included in a text )

VUl PHYSICS

book without encumbering it to the exclusion of important theo-
retical matter, and destroying its unity. We have therefore pre-
ferred to leave the choice of illustrative experiments largely in the
hands of the teacher, who may thus select them according to his
individuality, his equipment, and the circumstances and limita-
tions of his class and community.

I We have bnsed the argument wherever possible on the pupils'
1 experience, expecting this to be supplemented by the teacher
with lecture demonstrations and laboratory experiments, chosen in
accordance with the conditions which he has to meet and with
his own taste and judgment. But when a particular kind of ex-
perimental evidence is necessary to the argument, it has been used,
without manipulatory details and in uniform type with the other
subject matter.

In the third place, many of the old and familiar landmarks of
the elementary physics text do not appear in these pages. Among

f these may be mentioned the division of levers into classes; the
wedge; the classification of equilibrium as stable, unstable, and
neutral; specific gravity as distinguished from density; the elec-
trophorous and the electrostatic machine; the concave and con-
vex mirrors; multiple reflection; and the formulas concerned
with the radii of curvature of lenses. These have been omitted
because they seem of less interest and importance than the'
following new subjects which we have been able to introduce in
the space thus saved: The use of graphical methods and of
vectors; the discussion of efficiencies of engines, both prac
tical and theoretical; the relations among electrostatic charge,
current, and magnetic field; the meaning of harmony; the nature
of spectra; the reasons for the electromagnetic theory of light; and
the electron theory of matter. We also believe that the presenta-
tion of the subjects of rotary motion and of optical instruments
will be found much simpler and more satisfactory than those
usually given.
/ The problems are also an innovation. They include no
J cases of forces a, 6, and c, meeting at a point q, etc., but are,
. as far as possible, real, concrete cases, such as occur in actual
practice, and which every boy or girl ought to know how to meet.

PREFACE IX

( They also contain many of the subjects usually placed in the text
I and there explained; for example, the pulleys, distillation, and the
Wheatstone bridge. We hope that this form of problem will
Unterest the student, as most of them are problems in whose solu-
tion he can see some use.

Other devices for catching and holding the interest are the
questions and the suggestions to students at the end of each chap-
ter. We hope that these latter will be stimulating to the students
and serve as hints which will lead them to suggest for them-
selves other home experiments. Are not such experiments, clumsy
though they be, yet made with a genuine interest in finding
out something — in getting the answer from Nature herself — far
more useful than many that are made in some laboratories?

The illustrations are also a novelty. Great pains have been

'^(taken to have every picture a photograph of a real thing, for a

photograph is always more interesting than a woodcut. It is

believed that these will add much to the interest of the work.

We have been favored with the original photographs for
many of these illustrations, by the firms and individuals men-
tioned on page x, whom we wish to thank for their courtesy.

We also desire to express our thanks to Professor R. D.
Salisbury of the University of Chicago, Editor-in-Chief of the
Lake Science Series for many valuable suggestions, and to
Messrs. A. A. Knowlton of the Armour Institute of Technology,
J. H. Kimmons of the Austin, Chicago High School, and
C. Kirkpatrick of the High School, Seattle, Washington, for aid
in the reading of the proof.

Many of the line diagrams are new and have been designed
and executed with much thought and care, so as to present the
essential ideas without complication by unnecessary details.

That great difficulties are involved in the working out of a
method of instruction differing in principle from that in general
use must be apparent to every one. We know better than any one
else can that we have not produced a perfect book. This might
be approximated by the concerted action of all teachers of physics.
We therefore hope that members of the teaching fraternity will
regard the result of our work as a first approximation, and will

X PHYSICS

join with us in making a united effort to lift our subject up to
its proper place, and to inspire our young friends with an adequate
appreciation of its interest, its majesty, and its grandeur. To
this end we appeal to our colleagues to give us the benefit of their
experience by sending us suggestions and criticisms, which will be
gratefully received and carefully considered.

Charles Rtborg Mann,
George Ransom Twiss.

ACKNOWLEDGMENT OF ILLUSTRATIONS

^ Plate I. The Lake Shore and Michigan Southern Railway. Large

copies of this picture in color may be obtained for 50 cents, by

applying to Mr. A. J. Smith, General Passenger Agent, Cleveland,

Ohio.
Fig. 11. The Electric Vehicle Co., Hartford, Conn.
Figs. 16, 17, 18. The Eastman Kodak Co., Rochester, N. Y.
Fig. 19. Pawling, Harnischfeger & Co., Milwaukee, Wis.
Plate II, and Figs. 70, 156, 157, 158. The Niles-Bement-Pond Co.,

New York.
Fig. 31. The Manitou and Pike's Peak Railroad Co,, Manitou, Col.
Fig. 51. Crowe Bros., House Movers, Chicago, 111.
Plates III, IV, VI. The AUis-Chalmers Co., Milwaukee, Wis.
Fig. 61. The Bausch and Lomb Optical Co., Rochester, N. Y.
Figs. 65, 72. The Ingersoll-Sargeant Drill Co.. New York.
Fig. 74. The Chicago Bridge and Iron Works Co., Chicago, 111.
Fig. 77. The Century Co., New York.
Figs. 91, 92. The Whitlock Coil Pipe Co., Hartford, Conn.
Fig. 100. The Otto Gas Engme Co., Philadelphia, Pa.
Fig. 101. Mr. Alfred Stieglitz, New York.
Plate VII, and Figs. 102, 131, 132, 133, 141, 142, 143, 144, 147,

148, 151. The Westinghouse Electric Co., Pitlsburg, Pa.
"The Electric Spark in Nature," page 205. Mr. M. I' Anson,

Newark, N. J.
Fig. 162. The Electric Controller and Supply Co., Cleveland, Ohio.
Figs. 168, 169, 170. ' The Electric Storage Battery Co., Philadelphia,

Pa.
Plate VIII. The University of Chicago, Chicago, 111.
Figs. 236, 237. Wm. Scheidel & Co., Chicago, lU.

TABLE OF CONTENTS

PAGE

Introduction 11-14

CHAPTER I
Motion, Velocity, Acceleration —

Motion of a train — How velocity is measured — Units —
Graphical representation of velocity — Analytical repre-
sentation of velocity-^Slope — Changing velocity — Accel-
eration — Graphical and analytical representation of accel-
eration — Measurement of acceleration — Summary — Ques-
tions — Problems — Suggestions to students 15-32

CHAPTER II
Mass and Energy —

Production of acceleration —Acceleration and force — Dif-
ferent bodies having the same acceleration — Mass —
Masses compared by forces — Relation of force, mass, and
acceleration — Unit mass — Weight — Galileo*? experiment
— Weight and mass — Density — Work, force and distance —
Unit work — Energy, how measured — Efficiency — Kinetic
and potential energy — Newton's laws of motion —
Power — Engineering units — Summary — Questions —
Problems — Suggestions to students 33-56

CHAPTER III
Composition and Resolution of Motion —

Up grade — Composition of motions — Vectors — Motions at
right angles — Motions not at right angles — Vector solu-
tions — Analytical solution — Traveling crane — Resolution
of motions — Force vectors — Balanced forces — Mechanical
advantage — Summary — Questions — Suggestions to stu-
dents 57-72

6 CONTENTS

CHAPTER IV

PAGE

Moments —

How rotation is produced —Moment of force — The lever —
Work done by the lever — The lever principle — Parallel
forces — Weight and center of mass — Equilibrium — Sta-
bility, how measured — Determination of the center of
mass — Mechanical advantage of a composite machine —
The law of machines — The screw — The equal arm bal-
ance — Review — Summary — Questions — Problems — Sug-
gestions to students 73-96

CHAPTER V
Rotation —

Flywheels — Angular measurement and units — Correspond-
ence with linear measurements and units — Moment of in-
ertia and mass — Determination of moment of inertia —
Conditions for circular motion, centripetal force— Burst-
ing wheels — Distribution of mass — Moment of mass —
Railroad curves — Spinning tops — Summary — Questions —
Problems — Suggestions to students 97-111

CHAPTER VI
Fluids —

Pumps — Air has weight — Torricelli's experiment — Pascal's
experiment — Mercurial barometer — Characteristics of
fluids — Pascal's principle — Hydraulic machines — Free
level surface of a liquid — Gases — Air pump— Guericke
— Density of air — Theory of pumps — Archimedes' prin-
ciple — Flotation — Buoyancy — Determination of density
— Boyle and his law — Summary — Questions — Prob-
lems — Suggestions to students . 112-136

CHAPTER VII
Heat —

Heat and work — Thermometers — Temperature scale —
Gases — Change of volume at constant pressure — Change
of pressure at constant volume — Air thermometer — Ab-
solute temperature — Expansion of solids and liquids —
Heat quantity — Gram calorie — Specific heat — Steam —
Evaporation — Pressure and temperature of saturated
vapor — Boiling point — Superheated vapor — Critical tem-
perature — Humidity — Dew — Latent heat — Water and
climate — Summary — Questions — Problems — Suggestions
to students 137-157

CONTENTS 7

CHAPTER VIII

PAGE

Transfer op Heat —

Conduction and convection — Applications — Radiation —

Diffusion — Evaporation — Gaseous pressure — Effect of

heating — Kinetic hypothesis — Radiation — The ether —

Prevost's theory of exchanges — Absorption— Absorbing

power of water vapor — Radiation and absorption — Heat

and light — Summary^-Questions — Problems — Suggestions

to students 158-170

CHAPTER IX
Heat and Work —

Mechanical equivalent of heat — Gas is heated when com-
pressed — Gas cools when it expands and does work —
Liquid air — Cooling by evaporation — Manufactured ice —
The steam enghie — Work done by the steam — The pres-
sure-volume graph — Back pressure — Lower pressure at
exhaust — Condensers — Higher boiler pressure — Heat
energy consumed — Efficiency and temperature — Com-
parison of efficiencies — The triple expansion engine —
The gas engine — The steam turbine — Summary — Ques-
tions — Problems — Suggestions to students 171-189

CHAPTER X
Electricity —

Transmission of power — Generators — Early knowledge of
electricity — Gilbert — Electrification — Conductors and in-
sulators — Repulsion — Discharge — Electroscope — Both .
bodies equally charged — Two kinds of charge — Polariza- .
tion — Charging by influence — Charge on the outside of a
conductor — Coulomb's law — Leyden jar — Condensers — .
Operation of a condenser — Discharge of condenser is oscil-
latory — Lightning — Summary — Questions — Problems —
Suggestions to students 190-210

CHAPTER XI
Magnetism —

Lodestone and compass — Magnetic curves — Magnetic
field — Like poles — Unlike poles — Permeability — Magnetic
circuit — Earth's magnetism — Unit pole — Law of magnetic
force — Chief characteristics of magnets — Electric currents
— Voltaic cell — Electromagnetism — Magnetic field of the
current — Electromagnets — ^Telegraph — Relay — Grounded

CONTENTS

PAGE

wires — Electric bell — Galvanometers — A suggestive ex-
periment — Motors — Motor parts — From toy to practical
machine^ Ampere's theory of magnetism — Magnetic field
of moving charges — Energy of a magnetic system — Sum-
mary — Questions — Problems — Suggestions to students. . . 211-243

CHAPTER XII
Induced Currents —

Sources of current — Current and magnetic field — Faraday's
discovery — Current induced by a moving magnet — Num-
ber of lines of force changed — Currents induced by cur-
rents — Iron core — Laws of induced currents — ^The dynamo
principle — The dynamo — Magnetos — Alternating current
dynamos — The induction coil — The transformer — Alter-
nating current motors — The telephone — Summary —
Questions — Problems — Suggestions to students 244-264

CHAPTER XIII
The Electric Current at Work —

Pressure and current in the arc lamp — Current strength —
Resistance — Laws of resistance — Ohm's law — Ammeters
and voltmeters — Electric power — Watt meters — Arc light
plant — Incandescent lamps, parallel distribution — Incan-
descent light plant — Heating effects of the current — Joule's
law — Heat loss in transmission — Three wire system — Al-
ternating current transmission — Divided circuits — Shunts
— Arc lamp regulation — Lifting magnets — Voltaic cells —
Energy of the cell — Polarization of cells — The ion hypoth-
esis—Commercial cells — Electrolysis — Faraday's laws —
Electroplating — Storage batteries — ^^ Retrospect — Sum-
mary — Questions — Problems — Suggestions to students . . . 265-299

CHAPTER XIV
Wave Motion —

Water waves — Origin of waves — Characteristics of waves —
What waves tell us — Wave motion — Wave length —
Period — Phase — Velocity of propagation — Waves of sim-
ple shape — Complex waves — Waves of different shapes —
Stationary waves — Summary — Questions — Problems —
Suggestions to students 300-316

CONTENTS 9

CHAPTER XV

PAOB

Simple Harmonic Motion —

Uniform circular motion — Displacement and force — The
sine curve — Period, mass, and force constant — Pendu-
lum — Uses of the pendulum — The Foucault pendulum-
Summary — Questions — Problems — Suggestions to stu-
dents 317-327

CHAPTER XVI

Sound —

Sources of soimd — Soimd a wave motion — Soimd waves
longitudinal — Velocity of sound — Resonance — Noise —
The piano — Pitch — Musical intervals — Laws of strings —
Vibrating rods — Tuning forks — Organ pipes-^Air columns
as resonators — Intensity — Summary — Questions — Prob-
lems — Suggestions to students 328-341

CHAPTER XVII
The Musical Scale —

Development of the musical scale — The related triads —
The vibration numbers — The major scale — The complete
scale — The tempered scale — Standard pitch — Forced vi-
brations — The ear — Beats — Discord due to beats — Sum-
mary — Questions — Problems — Suggestions to students . . . 342-356

CHAPTER XVIII
Harmony and Discord —

Wave shape and tone quality — The vibrating flame — Mus-
ical tones complex — How musical tones are possible — Fun-
damental and overtones — Overtones of strings — Reson-
ators — How the ear perceives a complex tone — Related
tones — Chimes — Summary — Questions — Problems — Sug-
gestions to students 357-368

CHAPTER XIX
Light —

What does light do for us — Direction — Image by a pin .
hole — Image by a lens — The eye — How light is changed in
direction — Refraction — Index — How the lens forms the
image — Reflection — Diffuse reflection — Summary — Ques-
tions — Problems — Suggestions to students 369-383

10 CONTENTS

CHAPTER XX

PAGE

Optical Instruments —

Principal focus — Image of a point source — Construction of
the image — Lens angle — Size and distance of the image —
Virtual image — How the eye is focused — Spectacles — The
simple microscope — The camera — Stops — Spherical aber-
ration — The telescope — The concave lens — The opera
glass — The compound microscope — Magnification — Reso-
lution — Smnmary — Questions — Problems — Suggestions to
students ' 384-402

CHAPTER XXI

Color —

Newton's experiment — Interference fringes — Wave length
and color — Interference in white light — Dispersion — The
spectrum — Bright-line spectra — Measurement of disper-
sion — Achromatic lens — Spe.ctrum analysis — How the eye
perceives color — Mixing colors — Colors of ordinary ob-
jects — Paints and dyes-^Mixing pigments — Summary —
Questions — Problems — Suggestions to students 403-422

CHAPTER XXII
Velocity op Light —

. What we can learn from the velocity of light — Galileo's
method — Fizeau's method — The velocity determined —
Velocity of electric waves — Wireless telegraphy — Light
and electricity operate through the same medium — The
complete spectrum — Summary — Questions — Problems —
Suggestions to students 423-434

CHAPTER XXIII
Electrons —

Light waves start at an electrically charged particle — Other
such particles — Cathode rays — Action of magnet — deter-
mination of ~ — Comparison with the ion of electroly-
sis — The electron — Radioactivity — The X-rays — The na-
ture of white light — Conclusion 435-446

Index 447-453

PHYSICS

INTRODUCTION

It has been said that man made his sta,rt on the long road
toward enlightenment when he learned how to make a fire. For
many centuries, our ancestors groped at a snail's pace along this
road where we of the twentieth century are advancing by leaps
and bounds.

By slow and painful steps, prehistoric man learned to use fire
in order to keep himself warm, to cook his food, to get metals out
of their ores, and to forge them into rude tools and weapons of de-
fense. By means of signal fires on the hilltops, he sent his first
wireless messages across the valleys. The magnetic force of the
lodestone and the electric attraction of amber were known to the
ancients, and the fact that steam pressure can be made to produce
motion was known in the early centuries of our era. Why was it
that so many centuries elapsed before man learned to subdue these
forces of nature and make them do his will? Now we have the
steam engine, the electric dynamo and motor, the power printing
press, the power loom, the telephone, the wireless telegraph. By
means of these and countless other inventions, one man can do the
work of hundreds, the continents are linked together, darkness is
turned into light, time and space are vanquished.

We can best realize how important are these inventions when we
try to think how we should get on without them. And yet this great
development of miracle working machinery has come within the
space of three centuries, and the greater part of it within seventy-
five years 1

The stories of how these marvelous inventions came to be, of the
struggles of the men who brought them into being, and of the pa-
ll

12 PHYSICS

tient researches and brilliant discoveries of the men of science who
established the foundation principles upon which all these inven-
tions rest are among the most important and most interesting chap-
ters of history.

In the studies which follow, we shall endeavor to get an
understanding of some of these principles, to gain at least a slight
acquaintance with some of the great discoverers who formulated
them, and to get some insight into the kind of thinking and the
methods of experimentation by which their truth has been made
plain. Such studies are of interest not only to those who expect
to make practical use of them, but also to those who, in the pursuit
of a liberal education, wish to learn how to think clearly, to ex-
press themselves precisely, and to test their conclusions accurately,
as well as to get a properly balanced view of human life and ac-
tivity.

The principles of physics are most easily understood by the
beginner, and are also most interesting, when they are studied in
connection with his own experiences. For no one can live long in
this scientific age, surrounded as he is on all sides by the fruits of
discovery and invention, without having a large amount of experi-
ence with the forces of Nature and without obtaining therefrom a
large fund of general information.

A rapidly-moving railway train is certainly a familiar object to
every one. Even a small child would not have to be told that
Plate I is the picture of such a train. Moreover, we all know that
the locomotive causes the train to move, and that it can not do so
unless it has a fire in it. We are also familiar with the fact that
the locomotive must be supplied with water, and that in the boiler
this water is converted into steam, which somehow makes the big
driving wheels turn. That such an engine warns us of its approach
by means of a whistle and a bell, and that it lights its own path
in front of it at night by means of a brilliant headlight, are well-
known facts.

Now, although these and many other things about the locomo-
tive are matters of everyday knowledge to most of us, how many
of us can tell exactly how the steam makes the engine's driving
wheels turn? And why is steam used at all? Why are some loco-

INTRODUCTION 13

motives large while, others are small? How does the whistle
work, and how does its sound get to us? How is the headlight
made to send its light forw^ard on the tracks?

A ride in a steam or trolley car is one of the most common of
our experiences, and we all know that the car has many different
kinds of motion. Wherein do these motions differ? How are
speeds measured and compared with one another? What sort of
velocity has the car while it is starting or stopping? Why are
we thrown against the side of the car when it rounds a curve?
How is it that some engines can go faster than others?

The picture of the Twentieth Century Limited (Plate I) was
taken while the train was running at full speed. How do cameras
and lenses work?

We can obtain the answers to these and to other similar questions
without great difficulty, if we are willing to devote to the subject
some careful study and thought. When we have done this, w^e
shall find that the knowledge thus acquireti gives us a greater con-
trol over the forces of Nature, and that the training thus obtained
is of great service to us in everything we may wish to do.

CHAPTER I --X;.

MOTION, VELOCITY, ACCELERATION ' ' y

• •

1. The Motion of a Train. In order to find the answe^^fci

some of the questions just asked, let us suppose that a locomdtk'f "
stands with steam up, ready to make the run to the next station/
When it starts, we notice that at first it moves slowly, and that its
velocity gradually increases until it has attained **full speed," when
it runs for some time at a rate that is nearly constant. As the
next station is approached, the speed gradually decreases; and the
train comes to a full stop. What can we leam of its motion, of the
way in which it rounds curves, of how it gets up speed, and of
how it stops? How shall we describe and measure its velocity,
and how take accbujit of the energy that it must expend in order
to move its load?

2. How Velocity is Measnred. Since all motion implies
both distance and time, and since distances and times must be meas-
ured in order to be compared, it is necessary to have units of length
and of time in terms of which the measurements can be expressed.

The units adopted in all scientific work are
purely arbitrary, and are chosen simply for con- £~\ r~~]^,

venience. The unit of length is the centimeter, \ \ / / g
which is the one-hundredth part of the dis- \ V — / / J
tance between two lines on a certain bar of plati- / / V \ ^
num-iridium when the bar is at the temperature p / \ n j

of zero degrees Centigrade. This bar is care- fig. i

fully preserved at Paris, and is called the Inter- Standard Meter^
national Prototype Meter. The symbol for
centimeter is cm, and that for meter is m. The unit of time is
the SECOND, which is the one-eighty-six-thousand-four-hundredth
of the mean solar day. Its svmbol is sec.

Now if a train, moving uniformly, attains in one minute a

16 y *. tPHYSICS

distance of 150,00g cpi*from a given post in a certain direction,
then in one sepgft>3L*tfce change in its distance in that direction
from the post*. wiH *be -^^ of 150,000 or 2500 cm. Therefore it
travels at gti^e rate that its distance from the post changes 2500
cm eveiy'spftond. This rate of change of distance is called linear

VELOCITY. •

^ .. K'^without changing the direction of its motion, a body trav-
. if^ equal distances in equal times, no matter how small the time
• .in'tervals are taken, its velocity is uniform or constant. The unit
•^•'of velocity used in physics is the velocity of a body moving uni-
formly over one centimeter each second. If the body traverses
2 cm in each second, its velocity would be two units, or 2 centi-
meters per second, and so on; therefore, a unijoria velocity is
measured by the number of centimeters passed over in one second,

3. Comparison of Velocities. Let us now compare the veloc-
ity of our train with that of a fast freight which passes over a
distance of 90,000 cm from a given post in onfe minute. Its veloc-
ity is then ^V of 90,000 = 1500 cm per second, which is evidently
I of 2500, the velocity of the express. Similarly, a rifle ball that
passes over 240,000 cm in 3 seconds, has a velocity of 80,000 cm
per second. In all of these cases we obtain the number that
expresses the velocity by dividing the number of centimeters in
the distance by the number of seconds in which that distance is
traversed. Since the expression for velocity is thus obtained, an
appropriate sjrmbol for linear velocity is ~. Symbols made in
this way will be found very useful because they show at once how
a quantity like velocity is expressed in terms of the fundamental

. units.

4. The Analytical Method. Uniform velocity may therefore
be measured by dividing the distance passed over in a given
time by the number of seconds in that time. Since this expression
is rather cumbersome, it is more convenient to state it in an
abbreviated form by means of algebraic symbols. This is done
by letting v represent the linear velocity, I the number of cm in
the distance traversed, and t the number of sec in which the

MOTION, VELOCITY, ACCELERATION

distance is traversed. We may then write the expression

... Distance passed over in a ffiven time . . „

uniform velocity = — .r^ — -, — ^—^ i — : — " , .. m the form

•^ Number oi seconds m that time

(1)

This is the equation for uniform motion. This method of express-
ing relations by means of an algebraic equation is called the
ANALYTICAL METHOD. This method is extensively used in physics
and engineering, and has the advantage of great conciseness.

•9

—

■

_„.

Sgo

- -

f^-

- -

■ —

•-7o

—

" -

^5«

4 6 8
Time In Hours

4 6 8 10

Fig. 2. Variations of Temperature
During One Day

5. The Graphical Method. There is another very convenient
method by which relations of this kind are expressed. This
method is familiar to every-
body, since it is very gener-
ally employed to picture the
relative variations of two
quantities, both of which
are continuously changing
in value. Thus Fig. 2 repre-
sents the variations of tem-
jjerature during a day. The
time intervals are repre-
sented by horizontal dis-
tances, and the corresponding temperatures by vertical distances.
A single glance at the diagram tells us whether the range of
temperature on that day was large, when it was highest, when
lowest, and how hot or cold the air was. This method of present-
ing relations is called the graphical method.

Let us then apply this method to the train mentioned in Art. 2.
Since in this case the two quantities that vary are time and the
distance of the train from the given post, we must let one of the
quantities be represented by horizontal distances, and the other
by vertical distances. We may choose freely what scale to use,
i.e., how great a length on the diagram shall represent a sec or a
cm. In this case we shall get a drawing of convenient size if we
let 1 cm in the horizontal direction OX represent 1 sec, Fig. 3; and
1 cm in the vertical direction OY represent 2000 cm.

PHYSICS

If now we begin to consider the motion at the instant when
the front of the engine, going at the rate of 2500 ~, passes a cer-
tain post: then at that instant, since no time has elapsed and no
distance been passed over, i.e., the time is zero and the distance
also zero, the corresponding point on the diagram will lie at 0,
Fig. 3. At the end of one second the train is 2500 cm from the
post. Therefore the point that corresponds to this condition

must represent a time of 1 sec and
a distance of 2500 cm, and so
must be 1 cm from OY in the
direction OX, and 1.25 cm from
OX in the direction OY, To locate
this point we lay off 1 cm along
OX to a?!, and draw from x^ a
dotted line parallel to OY. We
then lay off 1.25 cm along OY to y^
and draw through y^ a dotted line
V parallel to OX, The intersection p^
— ^of these two dotted lines will then
be the point sought, since it is 1 cm
from OY and 1.25 cm from OX.
In like manner, at the end of the second second the train is
5000 cm from the post. So we lay off 2 cm along OX to the point
X2 to represent 2 sec, and 2.5 cm along OF to the point ^2* *o
represent 5000 cm. We then draw the dotted lines as shown in
the figure, and find the point p2, which therefore represents the
conditions at the end of 2 sec. Similarly, the point pg, distant
3 cm from OY and 3.75 cm from OX^ represents the conditions
at the end of the third second; and so on. Note carefully that
the line obtained does not represent the path of the train.

We next draw the straight lines Op^, p^ p^, pa Vzj ^tc. Is the
resulting line Opg straight? Do the points that represent the con-
dition of the train's motion at 0.5, 1.7, 2.2, 2.5 sec also lie on this
line? Is there on the line a point corresponding to every possible
instant of time? Does every such point also represent a distance
from the post? Does the time Opg completely represent the
motion of the train with respect to both distance and time?

Fig. 3.

1 \2 3

Uniform Velocity

MOTION, VELOCITY, ACCELERATION

Since we shall often use the graphical method, we shall need
to know the names of the lines and points. The two lines OX
and OF are called coordinate axes. The distances Ox^, Ox^y
Ox^j etc., are called abscissas. They may be measured from any
point on OF along a line parallel to OX; thus y^ pi, = Ox^,
2/2 P2i = 0^2> ^t^- The distances Oy^, Oy^y Oy^, etc., are called
ORDi nates, and may be measured from any point on OX along a
line parallel to OF. OX is called the axis of abscissas, and OY
the AXIS OF ORDINATES. is Called the origin of coordinates. The
line O/jg representing the relations considered is called a graph.

Fig. 4.
Slope Indicates Velocity

6. What the Slope Indicates. Let us now add to our diagram
graphs for two other trains, one of which is a fast freight F
traveling uniformly at the rate of
1500 ^, and the other an express E at the
very high speed of 3000^. The result
is sbo\\Ti in Fig. 4. In what respect are
the second and third graphs like A, the
first? Which graph has the steepest
SLOPE, or in other words which makes
the greatest angle with the axis of
abscissas? Would the graph for a slow
freight traveling at a rate less than
1500^ have a greater or a less slope
than that of the fast freight? Would the graph RB for a rifle ball
having a speed of 80,000 ^ make a greater or smaller angle with
the axis of abscissas than does that for the express? What charac-
teristic of the motion is indicated by the steepness of the slope?

It thus appears that in the graphical method of representation
the velocity is represented by the slope, while in the analytical

method (c/. Art. 4) it is measured by the ratio — . The slope and

the ratio — then serve the same purpose. But on the graph / is

represented by the vertical distance pn (Fig. 5), and t by the
horizontal distance On. Therefore the slope of the line Op may

be appropriately measured by the ratio ^. But pn is the side

PHYSICS

opposite the angle of slope, and On is the side adjacent to it
in the right triangle pOn; and in a right triangle this ratio of
the side opposite the angle to the side adja-
cent to it is called the tangent of the
ANGLE. It is clear that the tangent of a
given angle has always the same value
no matter what the size of the triangle is.
Thus, since the triangles pOn and pqm

. ., pn pm

are similar, f—-=^—.

On qm

Hence the appro-

priate measure of slope is the tangent of the
angle that the graph makes with the axis
^^°- ^' of abscissas.

Measurement of Slope

7. Increasing Velocity. Thus far we have considered the
motion of the train only when it is uniform. What now are the
characteristics of the motion
just after the engineer has
opened the throttle, so that
the train is getting up speed;
and what of the motion when
he has shut off the steam and
applied the brakes, so that
the train is slowing down?

Since, now, the velocity is
changing at every instant, it
can not be measured by the
distance traversed in one
second. Therefore the velocity at any instant is measured by
the distance which would be traversed in one second, provided that
throughout that second the rate were to continue the same as it
was at the given instant. Suppose now that the train starts from
rest, and that at the end of the first second it has gained a veloc-
ity of 50 ^, that at the end of the second second its velocity is
100 g^, and at the end of the third second 150 ^^; i.e., suppose
that the velocities at the end of successive seconds are as fol-
lows:

Fig. 6. Ready to Start

MOTION, VELOCITY, ACCELERATION 21

lec

sec

sec
5

cm
sec
250

300

100

350

150

400

200

etc.

Is the change of velocity for any one second the same as for any
other second, i.e., is the change of velocity constant? If during
the interval between the end of the eighth second and the end of
the twelfth the velocity changed uniformly from 400 to 600 ^,
what was the rate of change of velocity, i.e., the change of velocity for
any one second?

8. Decreasing Velocity. Again, let us suppose that when
the train is slowing down, its velocity changes in the first
second from 2500 to 2400 ^, and that at the ends of the successive
seconds the velocities are as follows:

sec

cm '

sec

cm
sec

2500

2000

2400

1900

2300

1800

2200

1700

2100

etc.

What is now the rate of change of velocity? Since this rate of
change is the ratio of the change of velocity to the time, it is
expressed as a number of ^^ per second. Thus if the rate of
change of velocity is such that 75 ~ is gained or lost each second,
then this rate of change is expressed as 75 centimeters per second
per second. It is customary to write this 75 ^^2-

9. The Name Given to Eate of Change of Velocity is Ac-
celeration. When the velocity is increasing, the acceleration is
positive; and when the velocity is decreasing the acceleration is
negative. When the acceleration is constant, as in the examples
just givQn, the motion is called uniformly accelerated motion.

22 PHYSICS

It is to be noted that the expression for linear acceleration is
obtained by dividing a number of units of velocity by a number of
units of time, ' Since velocity is length divided by time, it is plain
that acceleration is length divided by the square of time. Hence
the symbol for the unit of acceleration is ^j-

10. The Analytical Expression for Acceleration. The ana-
lytical expression for acceleration may be found as in Art. 8, except
that we now represent the related quantities by letters instead of
by numbers. Thus, if a represent the acceleration, V the velocity
at the end of a number of seconds denoted by t, and Vq the velocity

V — V
at the beginning of this time, then the acceleration is a = — - — ^.

This equation is simply the definition of acceleration written in
algebraic shorthand.

It is often necessary to find the change in velocity in terms of
the acceleration and the time. In order to do this, we multiply
both members of our equation by <, thus obtaining the result
V — Vq = at, i.e., the change of velocity is equal to a, the rate of
that change, multiplied by t, the time.

If we wish to find the value of the final velocity V when the
other quantities are known, we add Vq to both members of this
equation, which gives us

V = Vf^-{-at (2)

i.e., the final velocity is equal to the initial velocity 'plus the cJiange
in velocity.

11. Relation of Distances to Times. It will be interesting to
know what sort of lines we shall get if we plot graphs that repre-
sent the relations of distances to times while our train is starting
and stopping. In order to do this we must first know the distance
of the train from a given point at the end of each second.

At the beginning of the first second, since the train is at rest, the
velocity is zero: and the final velocity is this initial velocity plus
the change, or F = i^o + ^^> ^s stated in equation 2, Art. 10. Now
a, the acceleration, is 50 ^2 J hence the final velocity for the time 1
sec isF = + 50X 1 = 50 ^. Since the velocity begins atO and

MOTION, VELOCITY, ACCELERATION 23

ends at 50 ^, it must, during that first second, have all values
from zero up to 50 ^. Which of these values may we use in calcu-
lating the distance traversed in that second? Since according to
our supposition the velocity increases imiformly, the train will
traverse in a given time with the uniformly accelerated motion the
same distance that it would have traversed during that time with uni-
form motion at the average speed. The average or mean velocity,
then, is that which we must choose.

Since the velocity changes at a uniform rate, the average velocity
may easily be found by taking the arithmetical mean of the initial
and final velocities; and therefore, for the first second, if we represent

this mean velocity by v, we have v = — ^ — ^ ^^ s^-

Solving equation (1) for /, we have / = vt; and substituting, we
get / = 25 X 1 = 25 cm, the distance of the train from the starting
point at the end of the first second. Similarly for the time two
seconds (since for each time period we must consider the motion
from the beginning y in order to get the average velocity), the initial
velocity is 0; and the final velocity is F= + 50 X 2 = 100 ^-^.

Whence the average velocity 1;= = 50£I?; and the whole

distance traversed up to the end of the second second is again the
mean velocity multiplied by the time, i.e., I = vt =50X2 = 100 cm.
In like manner, for 3 sec, we get F = + 50 X 3 = 150 ~,

4- 1 ^0

and V = \ = 75^, therefore / = 0;^ = 75 X 3 =f 225 cm.

^ sec

By the same method of calculation, we find that the distances
for the first eight seconds are as follows :

sec

625

' 900

100

1225

225

1600

400

etc.

12. The Graph for Distance and Time. We now have the
data that we need, and can proceed to construct our graph. Let

PHYSICS

Fig. 7. Graph for Single Seconds

US choose our scales so that for the abscissas 1 cm represents 1 sec,
and for the ordinates 1 cm represents 100 cm. We locate the point
corresponding to each second (Fig. 7) and find them to be for
the beginning of the first second, pi for the end of the first second,

P2 for the end of the second

.„ . ^ second, p^ for the end of the

third second, and so on. If, as
before, we should connect the
points in succession by straight
lines, would the resulting line
be straight? Does the velocity
of our train change abruptly at
the end of each second or is it
increasing uniformly at every
instant? Does the broken line
connecting O, Pi,p2f etc., change
its slope at every point or only
at the points that we located?
Then does such a line properly represent the uniformly accelera-
ted motion of the train?

It ought now to be clear that the graph must change its slope
at every intermediate point as well as at the few points that we
located, ij it is to represent properly the uniformly increasing velocity
of the train.

If we should locate the points for the intermediate half seconds,
in addition to the points already placed, thus reducing our time
interval to 0.5 of its former value, and if we should connect all
the points successively as before, would the broken line thus ob-
tained more nearly fulfill the condition of changing its slope at
every point?

Suppose now that we were to reduce the time interval to 0.2 sec
and to plot the corresponding broken line (Fig. 8) ; would this line
approach more nearly than did the other to the line that would
represent exactly the uniformly increasing velocity of the train?
It must be clear that by continually diminishing our time intervals
we shall get broken lines that more and more nearly fulfill the con-
dition of changing slope at every point, and thus more and

MOTION, VELOCITY, ACCELERATION

Fia. 8
Graph for Fifths of a Second

more nearly approach to the graph that we want. It is obvious,

however, that, in a practical problem, it is useless either to ca/rry

the subdivision of the time inter-

vol beyond the point at which

the difference between the broken

line and a smooth curve is no

longer perceptiUe in the drawing,

or to use smaller time intervals

than we are able to measure by

means of the timepiece used in

making our observations.

In general when we wish to
make a graph that corresponds
to a series of observations, we
locate the points corresponding
to each of these observations, and
then draw the smooth curve that most nearly passes through all of
the points,

13. Slope of a Curved Graph. As long as \he line is a broken
one, the slope of the portion between any two consecutive points is
that of the straight line joining those points; but when we pass to
a graph that changes its slope at every point, it must be evident
that the slope at any point is approximately that of a straight line
joining the given point with a nearby point. The nearer we take
this point to the given one, the more nearly does the slope of the
straight line represent that of the curve.

Since, for this graph, the ordinates and the abscissas represent
respectively distances and corresponding times, just as they did in
the graphs for uniform motion, the slope at any point of this graph
must represent the velocity at the corresponding instant of tinier just
as it did in their case.

14. The Train is Stopping. In order to construct the graph
that will represent the relation between distance and time when
the train is slowing down, we must again calculate the distances
of the- train, at the ends of the successive seconds, from the j)oint

PHYSICS

at which the engineer applies the brakes. The velocity at
this instant is the initial velocity and is 2500 ^ (c/. Art. 8).
As the speed is decreasing, the acceleration is negative; and so
(ef. Art. 8), its value is a = - 100 ^2- Hence the final velocity

for the first second is F =
and the average velocity v

Vo-\- at = 2500 - 100 X 1 = 2400

2500 + 2400

= 2450^. Multiply.

ing the average velocity by the time as before, we get for the distance
traversed in the first second, I = vt = 2450 X 1 = 2450 cm. like-
wise for the second second we get V = 2300, v = 2400; so that
/ = 4800 cm. The values for succeeding seconds are as follows:
sec cm sec cm

5 11 250

1 2450 6 13 200

2 4800 7 15 050

3 7050 8 16 800

4 9200 etc. etc.

15. Graph for Negative Acceleration. Choosing scales such
that for the abscissas 1 cm represents 10 sec, and for the ordinates
1 cm represents 10,000 cm, and plotting precisely as before, we obtain

the graph shown in Fig. 9. In what way
is this graph for the case of negative ac-
celeration like that for the case of positive
acceleration (Fig. 8)? In what way do
these graphs differ? At the end of what
second does the train come to rest? As-
suming that the train then remains at
rest, add to the diagram the points corre-
sponding to the next five seconds. At
the end of what second does the graph be-
come parallel with the axis of abscissas?
What, then, is the slope at the end of the 25th second? At the end
of the 28th? the 30th? What velocity is represented by a slope of
zero?

16. The Entire Motion Bepresented. We have now the
graphs for the uniform motion of the express train going at full

ZO ZT> 36

Fig. 9
The Train is Stopping

MOTION, VELOCITY, ACCELERATION

speed, and for the uniformly accelerated motion with positive and
negative accelerations while getting up speed and slowing down.
In order that all of these motions may be represented by a single
diagram that will go on a page, a
smaller scale must be used. The
complete graph appears in Fig. 10
(1 cm = 30 sec, 1cm =30,000 cm).
Describe in succession the changes
of slope.

140

lOO

:io

17. Equations for TTniformly
Accelerated Motion. Passing now
to the anal3i;ical method of repre-
senting uniformly accelerated mo-
tion, let us develop an algebraic
expression that will generalize the
calculations of Art. 11 and Art. 14.
If Vq represent the initial velocity, a

. , 1 i' a *u *• T7 J.U Fig. 10. The Complete Graph

the acceleration, t the time, V the

final velocity, and / the distance, then by equation (2), Art. 10,

V = ^0 + at
Also, the average velocity, v is found by taking half the sum of the

6 9 lO

initial and final velocities; therefore, v =

^0 + (^0 + «0

On multiplying this average velocity by the time t to get the dis-
tance /. we have

l = Vot + — .

(3)

Equations (2) and (3) are the equations for uniformly accelerated
motion.

The laws of uniformly accelerated motion expressed by
these equations may be stated as follows:

1. The final velocity is equal to the initial velocity plus the
product of the acceleration and the time,

2. The total distance traversed is equal to the product of the
initial velocity and the time, plus half tJie product of the accelera-
tion and the square of the time.

PHYSICS

18. When the Moving Body Starts from Eest. In the cases
thus far considered the initial velocity was zero. On substituting
this value in the general expression, the term involving Vq vanishes

and the equations become V = at and /

— , which

express

the

relations when the moving body has started from rest.

It should not be forgotten that when the velocity is decreasing y a,
the acceleration , is negative.

19. Acceleration is Not Necessarily Uniform. Throughout
the preceding discussion we have assumed that the acceleration of
the train was constant. In reality the case is not quite so simple,
because the engineer at first puts on the steam pressure gradually,
and because the acceleration is diminished by the resistance of the
air, which increases very rapidly when the speed is increased. The
acceleration which we assumed to be uniform was the average
acceleration during the time considered.

20. Determination of Acceleration. The actual experiment of
determining acceleration is made by observing distances and

corresponding times, sub-
stituting their values in
equation (3), and solving
for a.

21. Translatory and
Eotary Motions. Thus
far we have considered
only motion in a straight
line. We are now ready
to define motion in general,
and to distinguish be-
tween translatory and ro-
tary motion.
Fig. 11. Translation and Rotation p^ y^^^^, j^ ^.^y ^^ ^^ j^

MOTION with reference to a given point when it is changing either its
distance or its direction from that point.

MOTION, VELOCITY, ACCELERATION 29

When a rigid body moves in such a way that all its p)oints describe
equal and parallel paths, its motion is called translation.

When the motion of a body is such that its points describe cir-
cumferences about some point or line, its motion is called rotation.
The point or the line about which the body rotates is called the
center or the axis of rotation. The planes in which the particles
move are all parallel to one another, and the axis is necessarily per-
pendicular to these parallel planes.

A sled going down a hill has translatory motion only, provided
there are no turns in the road; for then all of its points describe
equal and parallel paths. The same is true of a sail boat when it
is making a straight course. On the other hand, the buzz saw and
the grindstone are familiar examples of bodies that have rotary
motion only. Every point on the grindstone, for example, de-
scribes a circle about a point in the center of the axle on which the
stone is mounted. The centers of all the circles described by the
points lie on a line which is perpendicular to the planes of all the
circles. When an automobile is traveling along a straight road,
the body of the car has translatory motion only, while the wheels,
considered with respect to their axles, have rotary motion only;
but the wheels have both translation and rotation with reference
to a point on the road.

SUMxMARY

1. The units of length and of time are the centimeter and
the second. Their symbols are cm and sec.

2. Motion may be either translatory or rotary.

3. Linear velocity is the rate of change of distance in a given
direction.

4. Uniform linear velocity is measured by the distance traversed
in one second. Its symbol is g^.

5. Acceleration is the rate of change of velocity.

6. Uniform linear acceleration is measured by the change of
velocity in one second. Its symbol is ^.

7. Acceleration may be either positive or negative.

8. The distance traversed by a body having uniformly acceler-
ated motion is foimd by multiplying the average velocity by the time.

PHYSICS

9. The three methods of representing these relations are:

Analytical

(Equation 1)

V = Vo+ at
(Equation 2)

(Equation 3)

Verbal
Uniform or average velocity
equals distance divided by time.

With uniform acceleration, final
velocity equals initial velocity
plus acceleration multiplied by
time.

Distance traversed with uniform
acceleration equals initial veloc-
ity multiplied by time plus half
the acceleration multiplied by
time squared.

Time

QUESTIONS

1. Define the scientific unit of length, and give its symbol. Define
the unit of time and give its symbol.

2. Define the term, linear velocity. What is meant by a constant
linear velocity? What two things must be stated in order that the
velocity of a movmg body may be fully described?

3. How is the numerical value of a imiform velocity found? What
is the unit of velocity? What is its symbol?

4. ' Explain how to represent a constant linear velocity by the graph-
ical method. In connection with the diagram, point out and name the
co6rdinate axes, the coordinates, and the origin.

5. What characteristic of the motion of a body is shown by the
slope of the graph that represents it?

6. When the velocity of a moving body is changing, how can we
express numerically the velocity that it has at any instant?

7. Define acceleration, and illustrate by a numerical example.

8. When is an acceleration positive, and when negative? What is
meant by uniformly accelerated motion?

9. Draw the graphs that represent the relation of distance to time
for a positive and a negative acceleration. In these graphs, what does
the slope represent?

MOTION, VELOCITY, ACCELERATION 31

10. What changes of slope occur in the graph when the acceleration
is positive? When the acceleration is zero? When it is negative?

11. When a graph is curved, what line will represent approximately
the direction of its slope at any point?

12. Define motion, and distinguish between translatory and rotary
motion, illustrating by examples.

PROBLEMS ,

Note. 1 m = 100 cm = 39.37 inches.

1. A runner passes over 100 yards in 10 sec; what is his speed
cm p

sec'

2. What is the speed of a race horse that covers a mile in 2 min-

utesWin^^ (2) in 52?

3. What is the speed of an automobile that runs a mile in 55 sec.
W<^^ (2)i„|E?

4. Sound, at 0° Centigrade, travels 1090 feet in one sec. What is
the speed of sound in ^? How many seconds would it take to trav-
erse 1000 m?

5. What was the average speed of a railroad train that traveled
134 mUes in 115 minutes (1) in 2^|~? (2) in ^?

6. Express the velocity of 1 ^^ in ^*, and in ^.

7. A sled, started from rest and going down a hill of uniform slope,
traverses 900 cm from the starting point in 3 sec. What is the accelera-
tion, and what the final velocity?

8. A wagon starts down a hill with a velocity of 30 — and its

acceleration down-hill is 80 — ^? What is its velocity at the end of 5
sec? What is the total distance traversed in the same time?

9. A wheelman, starting from rest, had attained at the ends of
the first three seconds the following distances: 1 sec, 90 cm; 2 sec,
360 cm; 3 sec, 810 cm. Supposing the acceleration to remain con-
stant during that time, what is (1) the acceleration? (2) the velocity
at the end of 6 sec? (3) the distance traversed at the end of 5 sec?
(4) the distance traversed during the 5th sec? [For (4), subtract
the distance attained in 4 sec from that attained in 5 sec]

10. The results of experiments show that the acceleration of a body
allowed to fall freely is 980 -^j- (") Calculate the distances attained
by the falling body when given an initial velocity of 10 cm vertically
downward, making a table of distances and times up to 10 sec. (b)
Choosing a convenient scale, plot a graph representing the motion.

11. Calculate the velocities of the falling body for the times given
In problem 10.

32 PHYSICS

SUGGESTIONS TO THE STUDENTS

1. Which of you can make the longest list of the motions with
which you are familiar, classified under the headings: Uniform,
Uniformly Accelerated Positive, Uniformly Accelerated Negative,
Translatory, Rotary?

2. Mark your height on a door-post; measure it in inches and. in
centimeters. From these measurements can you find out how many
centimeters are contained in one inch?

3. In ypur debating society, choose for a question the following:
Resolved: That the general adoption of the metric system of weights
and measures is advisable. For data write to the National Bureau of
Standards, Washington, D. C.

4. Which of you can find out the most interesting facts about
Galileo and his knowledge of falling bodies?

CHAPTER II
MASS AND ENERGY

22. The Production of Acceleration. In the preceding chap-
ter, we attempted to get clear notions about uniform and accelerated
motions, without considering the factors upon which their production
and variation depend. What are the relations that determine
whether the motion shall be uniform or accelerated ? What relations
determine the amount of the acceleration? We can most easily find
the answers to these questions by again studying the train.

Let us first suppose that the train drawn by the engine consists of
six cars all alike. Let us also suppose that the engine, using its full
power, can impart to this train an acceleration of 50 |^. Now, if
this engine be replaced by a smaller one having less power, will the
acceleration that this smaller engine can impart to the same train be
greater or less than 50 ^? Must the engine that can impart to this

train an acceleration of 60 ^^^2 have greater or less power than the
first engine?

Those who can not answer these questions from observations
made upon the train itself, will readily answer them by inference from
similar cases. Thus, everybody knows that more force is required
for imparting to a ball a great velocity in a given time than
for imparting to it a small velocity in the same time, that two oarsmen
can impart to a boat a greater velocity in a given time than can one,
that greater effort is required by a bicyclist to attain a great velocity
in a given time than to attain a small velocity in the same time.
Observation and experience lead us habitually to associate a
greater acceleration with a greater effort or force.

23. Acceleration and Force. Although common experience
gives us this general information, it does not give us the specific
numerical relations. This information can be obtained only by
making careful measurements of the quantities involved.

PHYSICS

Thus, if we measure the pulls of different sized engines, having
different powers, and observe the corresponding accelerations,
arid if we make proper corrections for friction of the moving parts
and for air resistance, we shall find that the numbers representing
the pulls are directly proportional to the numbers representing
the accelerations imparted to the train.

Many experiments of this sort have been devised and carried
out in physical laboratories to test the validity of this conclusion,
and they all tend to establish the truth of the general principle that
when different accelerations are given to the same body, the ratio of
the numbers by which we express the forces to those by which we
express the corresponding accelerations is constant.

Fig. 12. Eight-Oared Shell

Another illustration will help to make this clear. When only
two of the crew of an eight-oared shell row, they can impart to the
boat a certain acceleration. After making proper allowance for the
increased resistance of the water and air, it will be found that when
four row, they can impart an acceleration twice as great, six an
acceleration three times as great, and so on.

24. Different Bodies Having the Same Acceleration. We have
thus far considered how the forces vary when different accelerations
are given to the same body. Let us now consider how the forces
vary when the same acceleration is given to different bodies.

If an acceleration of say 50 ^ can be given by a certain engine
to a train of five empty cars, must the engine that can give the
same acceleration to a train of ten similar cars be more or less
powerful than the first? Again, if the same acceleration is to be

MASS AND ENERGY

given to the train of five cars loaded with passengers, can the same
engine do the work?

Common experience again gives us qualitative answers; for
everybody knows that an engine that can easily move a short
train may fail to move a long one, so that another engine must
be added. Likewise it re-
quires greater effort on the
part of a bowler to give a
large ball a certain velocity
than to give a small ball
the same velocity in a given
time; and it requires the
efforts of more oarsmen to
give a certain acceleration to
a big boat than to a little one.
The student can recall many
similar facts from his daily
observation.

It appears, then, that the more we increase the size of a body,
the substance remaining the same, the greater is the force required
to give it a certain acceleration.

Fig. 13. Small Engine: Short Train

25. Mass. In order to get quantitative relations, experiment
is necessary. If we measure the pull of an engine when it is im-
parting an acceleration of 50 ^ to a train of five empty cars, and
again when it is imparting the same acceleration to a train of ten
similar cars, we shall find that the pull in the second case is twice that
in the first. Likewise we shall find that the pull for a train of fifteen
6ars is three times that for five cars, and so on; i.e., when the
acceleration is the same, the numbers representing the forces are
directly proportional to the corresponding numbers of cars.

The matter appears very simple as long as the cars are empty
and all alike. But although we know from experience that more
force is required to impart a given acceleration to a loaded
train than to an empty one, yet it is impossible to determine how
much force, until we have adopted a means of comparing the loaded
train with the empty one.

36 PHYSICS

These differences in the make-up of the trains, whether in the
number of cars or in the load, are differences in mass.

26. Masses Compared by Forces. It is easy to see that when
two trains consist of precisely similar cars, all of them empty,
the train of ten cars has twice the mass of the train of five cars, be-
cause it is made up of just twice as many units of the same kind.
But it has just been shown that to impart a certain acceleration to a
train of ten empty cars the force is twice as great as that for a train
of five empty cars; so that we may compare the masses of the two
trains not only by the numbers of cars, but also by the forces re-
quired to give them the same acceleration.

When the differences in the trains are differences in the loads,
we can not compare their masses by comparing the number of
cars, because the units are not alike. Therefore we must resort
to the other method, that of comparing the masses by the forces
that can impart to them the same acceleration. This method is
applicable to all bodies, whether composed of like or unlike kinds
of matter. Therefore, in general, tvx) masses are equal when, under
the same conditions, equal forces can impart to them equal accelera-
tions.

Applying this method to the cars, it appears that when an engine
can give two empty cars the same acceleration that it can give to a
single loaded car, the combined mass of the single car and its load
is equal to that of two empty cars ; and therefore in this case the mass
of the load is equal to the mass of one of the cars.

For a given acceleration, then, since the masses are equal when
the forces are equal, it follows that if one of the masses be doubled,
the corresponding force is doubled ; if the mass is made thrt^ times
as great, the force is also three times as great; and so on. In general,
then, when the acceleration is constant, the forces are proportional
to the masses.

27. Force, Mass, Acceleration. Since we have shown in the
preceding paragraphs that when the mass is constant, the force
varies directly as the acceleration, and also that when the acceleration
is constant, the force varies directly as the mass, it follows that, in

MASS AND ENERGY

general, the force must vary directly as the product of the mass and
the acceleration. If we choose our units of force appropriately,
and if we let / represent the force, m the mass, and a the acceleration,
we may write

/ = ma (4)

This equation defines the force in terms of mass and acceleration.

In connection with this equation, it is to be noted that if m
and / are constant, a must be constant also; i.e., if a body be
acted on by a single or an unbalanced constant force, its motion
will be uniformly accelerated.

Besides magnitude, every force has two other characteristics,
namely, its direction, and its point of application. When these
three characteristics are specified, the force is fully described.

28. Unit Mass. Thus far we have taken an empty car as
the unit of mass; but it is manifest that accurate measurement
necessitates the establishment of a
unit that is fixed and at the same
time more convenient. Therefore,
just as we have a standard of
length, the meter, we have also a
standard of mass. The interna-
tional STANDARD OF MASS is a cer-
tain piece of platinum which is
carefully preserved at Paris along
with the standard meter, and is
called the kilogram. The unit of
mass employed by all scientists is
the gram, which is the one-thou- fio. 14.
sandth part of the mass of the
standard kilogram. The abbreviation for gram is gm. To express
very large masses, the kilogram is a more convenient unit. The
abbreviation for kilogram is Kg.

Smce we are now able to express both mass and acceleration in
terms of grams, centimeters, and seconds, we may also express force
in terms of these same fundamental units. Thus in the equation

Standard Kilogram:
Actual Size

38 PHYSICS

/ = ma, if we substitute m = 1 gm, and « = 1 ^2> ^'^ obtain
/ = 1 X 1 = 1 ^^^> i^7^ic^ defines the scientific unit of force as thai
force which can impart an acceleration of one centimeter per second
per second to a mass of one gram. This unit of force is called the
DYNE. Note that the number of units of force is obtained by
multiplying together the numbers representing mass and accelera-
tion. This operation gives us gm x ^2- Hence the symbol for
the dyne is ^^.

V—v.

By definition (c/. Art. 10), a = — - — ^, therefore f ^ ma =
The product m{V — v^), or mass X change of veloc-

•miV-Vo)

711 ( V — V ^

ity, is called change of momentum; — ^^ — - — — is therefore the

rate of change of momentum; and since it is equal numerically to
may it is also a measure of the force to which it corresponds.

29. Weight. Since we have learned to state the relation
of force to mass and acceleration, we are in a position to get
some definite ideas concerning a subject about which there are
many common misconceptions. We all learned in early childhood
that bodies, including ourselves, fall to the earth when unsup-
ported. We are accustomed to associate this motion with a force
called gravity, which we conceive acts so as to attract all bodies
toward the earth. The attraction between the earth and any par-
ticular body is called its weight, and tends to give the body an
acceleration vertically downward. This fact is also a familiar one,
for everybody knows that a body falling from a great height
acquires a greater velocity than does one falling from a less height.
It is our knowledge of this fact, acquired from very early expe-
rience, that impels us to avoid a high fall.

Now, what is the relation between the weights of bodies and
their masses? Equation (4) will give us the answer. Thus^
the weight /, in dynes, of any body whose mass is m, is equal to
this mass multiplied by a, the acceleration that this weight will
give it if it is allowed to fall freely; i.e., / = ma. Similarly, the
weight f of any other body having a mass m', and receiving an
acceleration a\ is /' = m'a'. In order to find the ratio of the two

MASS AND ENERGY

weights in terms of their corresponding masses, we must divide one
of these equations by the other, thus: j = . We therefore see

TTia

that if both bodies have the same acceleration when falling
freely, i.e., if o' = o, then their weights are proportional to their
masses.

30. Gktlileo's Experiment. The question to be answered
now is, When two bodies have different masses, does the attrac-
tion of the earth give them equal accelerations? From the time
of Aristotle to the end of the
sixteenth century, this was a
much disputed question. Aris-
totle (384-322 B.C.) taught
that if t\/o bodies of unequal
.mass were dropped from the
same height at the same in-
stant, the heavier body would
reach the earth first; and his
followers defended this opinion
by his authority and by argu-
ments based upon what they
thought ought to be the nature
of things. Galileo (1564-1642
A.D.) was the first to recognize
that svxih a dispute can be set-
tled only by experiment. Ac-
cordingly, about the year 1590,
he performed the experiment
of dropping at the same instant
a small cannon ball and a large
bomb from the top of the Leaning Tower of Pisa. They reached
the ground at very nearly the same instant; so he came to the con-
clusion that if it were not for the resistance of the air, they would
have fallen in exactly the same time. The fact still remained, how-
ever, that a body with a large surface in proportion to its mass,
such as a feather, was known to fall very much more slowly than

Fig. 16. Leaning Toweb op Pisa

40 PHYSICS

a piece of metal. After the invention of the air pump, in 1660,
it became possible to settle the dispute finally. This was done
by showing that when a feather and a coin were dropped simul-
taneously in a long tube from which the air had been pumped,
they fell side by side and reached the bottom at the same time.

31. The Belation between Weight and Mass. Reasoning from
these experiments by means of equation (3), Art. 17, it follows
that the accelerations of all freely falling bodies are equal. For
if two bodies fall simultaneously through a distance I in time <,
then for the first, since the weight and hence the acceleration is

constant, I = -—; and likewise for the second, V = —^, But

since the distance is the same for each, as is also the time, I = V
and t = t', whence a = a'.

Thus it has been proved that at any given place, the accelera-
tion due to the earth's attraction is the same for all bodies and that
therefore, so long as they are compared at the same place, tlie
weights of all bodies are proportional to their masses,

Galileo, wishing to prove this statement with greater accuracy,
devised experiments with pendulums of different mass. These
experiments verified more accurately the same conclusion. Re-
peated with greater refinement by Sir Isaac Newton and others,
they have given convincing evidence of the truth of this statement.

From what has just been stated, it follows that v)e can cmtt-
pare vmsses by comparing their weights. This is the method in
common use; but it must be noted that, since the attraction of
the earth for a given body is different at different places, the
weights of the two masses that are to be compared must in gen-
eral be determined at the same place. For the comparison of
masses by means of their weights, the equal arm balance is gen-
erally used.

32. Density. In connection with the masses of different bodies,
we have seen that bodies having equal volume may differ greatly
in mass. Thus, one cubic centimeter of lead has a much greater
mass than has one cubic centimeter of water; while the latter has
a greater mass than has one cubic centimeter of wood. The

MASS AND ENERGY 41

appropriate measure of the density of any substance is the mass
in unit volume at a temperature of zero degrees Centigrade. Thus,
if the mass of a specimen of a certain kind of glass is found to be
25 gm, and its volume 10 cm', the average density of the glass is
. 1 of 25, or 2 . 5 grams per cubic centimeter.

If D represent the density, m the mass, and V the volume,
these relations are stated analytically by the equation

D = -

As defined by this equation, the density of a substance is its
mass per unit volume. The unit of density is one gram per cubic
centimeter, and its symbol is ^^, Since the gram was intended to
be the mass of 1 cm' of water, and since it is so, very nearly, the
number of cm' in the volume of a quantity of water is the same
as the number of gm in its mass. The density of water, there-
fore, may be taken as 1 ^^.

33, Work. •, In Chapter I we have studied the motion of a rail-
road train and seen how that motion is produced by the engine.
Why does the engine move at allf Must more steam be used to move
a train of large mass than to move a train of small mass? Must
more steam be used to move a given train over a long distance
than over a short one? Other conditions being the same, does
it require a larger amount of coal to generate a larger amount of
steam? The student probably knows the answers to these ques- p
tions, and also in a general way that, other things being equal, the
amount of coal required is proportional to the amount of work to
be done. Since most kinds of work, like that done by the locomotive,
consist in putting bodies into motion, and in maintaining tlieir
motions in opposition to resistances of some sort, and since some-
body always has to pay for getting work done, it becomes
necessary to know definitely just what an amount of work depends
on, and to have a unit in terms of which all kinds of work may be
measured.

34. Work, Force, Distance. If an engine or a horse or a
man is doing any kind of work it is evident that, other thuigs being

PHYSICS

Fig. 16. Plowing
Work is proportional to force and to distance.

equal, the amount of work done is directly proportional to the push
or pull, i.e., to the force exerted by the agent that does the work.
Thus, if each of two engines pulls its train on a straight and level
track for the distance of a mile, and if the second engine has to pull
with twice the force of the first, it is clear that the second engine

must do twice the
work that the first
does. Again, suppose
that the second en-
gine pulls for . one
mile, and then con-
tinues to pull with
the same force for
another mile, it must
again be clear that in
pulling the train two
miles it does twice
the work that it did in pulling it one mile. It follows, then,
that if the second engine, exerting twice the force of the first, and
hence doing twice as much work per mile as does the first, should
continue pulling with this force through a distance of two miles,
it would do four times as much work as the first engine did in
pulling its train one mile.

Since, then, the amount of work done by an agent is directly
proportional to the force and also to the distance through which
the agent acts, and since the amount of work depends on these two
factors only, it follows that when the units are properly chosen,
the measure of the work done is the jyroduct of the numbers repre-
senting the farce and the distance. In symbols, if / represent the
force of the agent, and I the distance through which it acts, and if
W represent the work done, then

W = fl. (5)

This is the equation for work.

35. TTnitWork. The unit in terms of which work is meas-
ured may easily be defined with the help of the equation W = //,
for if / = 1 dyne and 1=1 cm, we have W =1x1 = 1.

MASS AND ENERGY

Therefore, since the equation gives unity for the work when
the force is one unit and the distance one unit, it is most con-
venient to define the unit of work as the amount of work that is
done when a force of one dyne acts through one centimeter. This
is the unit of work adopted by physicists, and it is called the erg.
Since the symbol for the dyne is ^^^, and since the number
of ergs is obtained by multiplying together the number of dynes
and the number of centimeters, it follows that the symbol for the
ergis«^Xcm,orSMm?.

36; Unergy. We now come to the question of the relation
betweeWlhe^ amount of coal burned and the amount of work done.
It is generally recognized that
a water wheel, in order to
move machinery continuous-
ly, must be continuously sup-
plied with water, which must
be allowed to fall upon it
from a higher level; that a
windmill will not continue
to pump water unless the
wind continues to blow
against its blades, that a
horse or a man can not con-
tinue to do work unless he
regularly consumes food, and
that an engine of any sort
must continuously consume

coal in order that the steam may be kept up at the necessary pres-
sure while it is doing its work.

For centuries the most careful thought of philosophers and the
greatest genius of inventors were employed in trying to think out
and construct some device for obtaining perpetual motion, i.e., a
device which would continue to move indefinitely without a con-
tinuous external supply of energy. Since every such attempt has
been unsuccessful, scientists have become convinced that a per-
petual MOTION MACHINE is impossible. Thus, if any machine be

Fig. 17. Haying
A man can not work unless he consumes food.

PHYSICS

at rest, it can not start itself; and if it be in motion, the greater
the friction of its moving parts the sooner will it stop. If it be
harnessed to other bodies and made to do work in moving them,

it will come to rest all the sooner. It
can be made to work continuously
only by supplying it continuously with
ENERGY from some external source.
Energy y then, represents ability to do
work. In the case of the water wheel
the energy is derived from the motion
of a mass of water; in the case of the
windmill, from the motion of a mass of
air; in the case of the horse or man,
from the consumption of a quantity of
food; and in the case of a steam or gas
engine, from the consumption of a
quantity of fuel.

Thus it becomes evident that to
do work energy must be expended, and that to store up this energy
work o{ some sort must have been done. Now, many careful
experiments with all forms of energy have shown that a given
amount of energy always corresponds to the same amount of
work, whether that energy be expended in doing the work, or
the work be done in storing the energy.

Fig. 18. The Windmill

It will not go when there is no

wind.

37. Energy Measured by Work. Since the energy of a body
is equivalent to the work it can do, and also to the amount that had
to be done an it in order to impart the energy to it, we may measure
this energy by measuring either of these amounts of work. Some-
times one of these methods is more convenient, sometimes the other.
For example, let us consider the energy necessary to run an eight-
day clock. Such a clock is usually operated by a spiral spring, or
by a weight which is raised by winding up the cord upon which it
hangs. Suppose that the weight has a mass of 5000 gm. Then,
since / = ina, the force with which it pulls on the cord is ma,
or 5000 gm multiplied by the acceleration that it would have if
allowed to fall freely in consequence of the earth's attraction. This

MASS AND ENERGY

acceleration, which we have learned is the same for all bodies, is
found by experiment to be, at sea level and in the latitude of New
York, 980^. Therefore, the force with which we must pull in
order to lift this mass is 5000 X 980 = 4,900,000 dynes. To avoid
the repetition of zeros it is convenient to write this 49 X 10^.

If the distance through which the mass is lifted is 100 cm, then
from Art. 34, W = fl = 4 900,000 X 100 = 49 X 10^ ergs. Since
this is the amount of work done in
winding up the clock, it represents
the energy stored in the lifted weight.
Likewise when the weight descends,
it does work in running the clock and
this work is again f X I = 5000 X
980 X 100 = 49 X 10^ ergs. Since
this is the work done by the energy
stored in the lifted weight, it also is
a measure of that energy. Thus, in
general, if we can measure or com-
pute the work done on a body in
imparting energy to it, or the work
that it does when it parts with its en-
ergy, we can determine the amount of
energy that it had.

It will be noted that in the exam-
ple just given a small amount of use-
less WORK, done in overcoming fric-
tion while winding the clock, was
neglected. In every case in which
energy is transformed or transferred some of this useless work is
done. If the amount of useless work is at all comparable with that
of the useful work, allowance must be made for it. The ratio of the
useful work done to the total amount of energy expended is called
the EFFICIENCY of the machine by which the transformation or
transference i$ accomplished.

Suppose now that in the clock just considered we were to replace
the weight by a spiral spring. How much energy must the spring
have when wound up in order that it may be able to run the clock

Fig. 19. Hoisting Coal
Work equals force times distance.

PHYSICS

for as long a time as the weight ran it? How much work would
have to be done in winding up the spring?

38. Energy is Potential or Kinetic. In the cases that we
have considered, energy has been stored in a lifted weight, a

coiled spring, unbumed coal,
and unconsumed food. En-
ergy of this kind that a body
has because of its position or
internal condition, so that it
tends to move and do work, is
called potential energy. In
the case -of the windmill or
the water wheel, the energy
of the air or water is due to
the fact that it is in motion.
Likewise a base ball or can-
non ball does work while it
is being stopped. Hence it
also possesses energy; and it
must be quite clear that it
has this energy because of its
motion. The energy that a moving body has because of its motion
is called kinetic energy.

'/I

.**

Fig. 20. Pile Driver

The weight has potential energjr when it is

raised, kinetic when it strikes.

39. The Kinetic Energy of a moving body being evidently
due to its mass and velocity, it is often more convenient to measure
it in terms of these quantities than in terms of work done. This
may readily be done with the help of equations (3), (4), and (5).
Thus W = fl, in which / is the average force used in imparting
to the moving mass its velocity, and I the distance through which
this force acted. Also this force / = ma, in which m is the mass
of the body, and a its acceleration while acquiring its full velocity.
Therefore the work done in giving the body its kinetic energy is
W= fl = mal. Again, by equation (3), / = ^af, and by equation
(2) V = at, in which I is the distance traversed in the time t while
acquiring the velocity V with an acceleration a. Since we do not

MASS AND ENERGY 47

care to know the time t, we may eliminate it by substitution. Thus,

V P

from (2) / = — ; then f = —j. Substituting this value for t in

equation (3), we have I = -^ = — . Finally, by substituting this
value for I in the equation W = mat,, we find that the energy is
e = TF = — ^ — . Simplifying, we have

c == — - ergs. (6)

Since the symbol for mass is gm, and that for velocity is ^,
the symbol for kinetic energy is ^^^^ . Note that this is the same
symbol as that for the unit of work, as it should be, because en-
ergy is measured by work.

The advantage of deriving the equation e = --r— is mani-
fest when we apply it to the case of throwing a ball or firing a shot.
For while it would be very difficult to measure t and a correspond-
, ing to the distance I through which the force of the hand or the
powder was exerted, it is not so very difficult to measure F. Hence
it is often desirable to have an equation in which a and t are not in-
volved. In many cases, however, the kinetic energy of a body can
be measured with convenience by the work that it does when it gives
up its energy in stopping,

40. Newton's Laws of Motion. From all that has been said
it must be apparent that a body can not of itself start, or stop,
or otherwise change either the rat6 or the direction of its motion.
This fact is often expressed by saying that every body has inertia.

The relations of the phenomena with which we have become
familiar in this chapter were described tersely by Sir Isaac Newton
in the following statements, which first appeared in his celebrated
Principia in 1687. They are known as Newton's Laws of Motion.

1. Every body continues in its state of rest or of uniform motion
in a straight line, except in so far as it is compelled by force to
change that state.

2. Change of motion is proportional to the force impressed

48 PHYSICS

and takes place in the direction of the straight line in which the
force acts.

3. To every action, there is always an equal and contrary re-
action, or the mutual actions of any two bodies are always equal
and oppositely directed.

41. Illustrations. All these laws are illustrated in the train.
The train can not start unless pulled by the engine, which exerts
force upon it, i.e., imparts energy to it. Once started, the train
can not stop itself. If brought to rest it gives up its energy in over-
coming the resistance of the air, the friction of the moving parts,
and the friction of the brakes when they are applied to the wheels.
If there were no such resistances, the train, once set in motion
with a certain velocity, would continue to move without change
either of speed or direction. Again, while starting, if there were
no friction, and if a constant force were applied, there would be an
increase of velocity the same for each second, i.e., the rate of change
of motion, as measured by the product of the mass and the accelera-
tion, is proportional to the force; and it is in the direction of the
straight track along which the engine pulls.

When the brakes are applied, their force, and therefore the
corresponding acceleration, is in a direction opposite to that of the
motion; and if this force remains constant, there is a decrease of
velocity the same for each second. Here again the total change of
motion is proportional to the force impressed and takes place in
the direction in which this force acts.

But why is it that the train when under full head of steam does
not continue with uniformly accelerated motion, and therefore
increase its speed indefinitely, instead of reaching a certain speed,
which it can not surpass? The answer is that the resistance of the
air increases very rapidly, and therefore the engine soon has to use
all its energy against air resistance and internal friction ; so that there
is no excess left to do the work of increasing the motion of the train.
The total external force opposed to the motion of the train is
exactly equal to the total external force urging it forward; and
therefore the result is the same as if no force were a^cting at all —
namely uniform motion in a straight line.

MASS AND ENERGY

Fig. 21. Diving

The boat has an acceleration in the opposite

direction.

Now where are we to look for the application of the third law?
We have seen that the engine can not move the train if the driving
ivheels sHp; therefore it appears that the force of the engine is
applied at the place where the drivers bear upon the track. The
engine, then, tends to push the track backward, and would do so
if the track were free to
move. But the track is
made fast to the earth,
and therefore the engine
tends to push the whole
earth backward. The
force of the engine, the
action, is equal to ma,
i.e., to the total mass of
the engine and train mul-
tiplied by the accelera-
tion that they acquire.
Also the resistance of
the earth, the reaction,
is equal to mV, i.e., to the mass of the whole earth multi-
plied by the acceleration that it receives. This force and accel-
eration are oppositely directed with respect to those of the train.
Why does not the earth move? The answer is that it does move,
but so little that the motion is imperceptible. This will easily
be understood when we remember that the two forces are equal, i.e.,
mV= ma. But, dividing both members of the equation by m'a, so

as to get the ratio of the accelerations, we have —7- = -7-, whence
° ma m'a

— — — ;, which tells us that when the forces are equal the accelera-
a m'

tions are inversely as the masses. Since the mass of the earth is

very large compared with that of the train, it is evident that the

acceleration of the earth must be very small. If the masses were

more nearly equal, the accelerations would be more nearly equal.

Every one knows that when he dives or jumps from a small boat

it has a perceptible acceleration in the direction opposite to that in

which he jumps; and if he jumps from a larger boat, the accelera-

50 PHYSICS

tion of the boat is smaller; while if he jumps from a big ship, the
acceleration of the ship is imperceptible. So it is with the engine
and the earth.

42. Kate of Boing Work. In connection with work and
energy there remains another important question to be considered.
How are we to measure the rate at which energy is supplied; or,
what comes to the same thing, how are we to measure the rate
at which work is done? With the units we have adopted this is
very simple. We have only to calculate the number of ergs of
work done per second. Thus, if 120,000,000 ergs are done in 60
sec, then in 1 sec there will be perfonned one sixtieth of 120,000,000
or 2,000,000 ergs. The rate, then, is 2,000,000 ^^|. The rate
at which any agent does work is called its power or activity, and
the power is measured by the work that it can do in one second ; i.e.

Power = -^^ .
seconds

43. Engineering Units. For measuring force, work, energy
and power, engineers use a system of units based on the pound
weight, the foot, and the second. These units are not nearly so
convenient as those based upon the centimeter, the gram, and the
second; but since they are so widely used in engineering practice
they are here described for reference. Those students who expect
to prepare themselves for engineering, should master these defini-
tions and be able to apply them in numerical problems.

Since the foot is equal to 30.48 cm, the numerical value of the

980 ft

acceleration of gravity in this system is q , or 32.2 — g (nearly).

oU.4o sec

This quantity is usually denoted by g.

Instead of deriving their unit of force from a unit of mass and
a unit of acceleration, as the physicists do, engineers use as their
unit of force the weight of a pound mass at sea level and in the lati-
tude of New York, and call it the pound-force.

Whenever in an engineering equation the mass of a body ap-
pears — as in the case of kinetic energy — it should be noted that we
must eliminate it from this equation with the help of equation (4),

MASS AND ENERGY 51

which expresses the relation between the mass of a body and its
weight. Thus, / = ma, whence m = — . But / is expressed in

pounds-weight, and o, the acceleration in this case, is 32.2 — -^;

, . ., p , 1 pounds-weight of the body,
therefore the mass of a body m = ^^-^ ~

which expression must be substituted for the mass in the given
equation.

The amount of work done or of energy expended when a jxmnd-
force is exerted through the distance of 1 foot, is called one foot-pound.
By equation (5), Art. 34:

W (in f oot-pounds)\ = // = pounds-force X feet.

To get the measure of the kinetic energy of a body, in terms

of its weight and velocity, we must resort to the equation,

mV'^
e = — ^— . Since in engineers' units the mass m of the body is

~ ^^ — , and since the velocity V is expressed in feet per

second, the equation becomes^

pounds-weight (feet per second)^
e = - X 2 ' ^^

.... , . pounds-weight X (feet per second)'
e (m foot-pounds) = ^ 32 2 X 2

The engineers unit for power or activity is tlte horse-power,
which is the rate at which work is done or energy expended when
550 foot-pounds of work are done in ea^h second. Hence
-J. _ pounds-force X feet

^ 550 X seconds

One horse-power is found to be equal to 746 X 10^ ^^.

On the continent of Europe, engineers use a system of units
based upon the kilogram, the meter, and the second. These units
are defined or derived in a similar manner.

Thus, the kilogram-meter is the work done, or energy
expended, when a force that is equal to the weight of a kilogram
mass is exerted through the distance of 1 meter. Hence,

W (in kilogram-meters) = fl = kilograms-force X meters.

52 PHYSICS

One kilogram-meter equals 980 X lO"* ergs.

Since g, the acceleration of a freely falling body, expressed in
meters and seconds, is 9.8 ^, the equation for kinetic energji in
kilogram-meters is

_ mV^_ kilograms-weight (meters per second)'

^ - "2" - g ~ ^ ] 2 ' ""^

,. , ., , V kilofframs-weiffht X (meters per sec*. )*
e (m kilogram-meters) = .

To solve problems in which the relations are expressed by these
equations, it is necessary only to substitute the known values for
the quantities represented in the equations, each expressed in its
appropriate units; and then the unknown quantities can be found,
provided, of course, that in the statement of the problem, one equa-
tion can be formed for each of the unknown quantities.

SUMMARY

1. To describe a force completely we must state: 1, its point
of application ; 2, its direction ; 3, its magnitude.

2. Two bodies are said to have equal masses if equal forces give
them equal accelerations.

3. The unit of mass is the gram, and its symbol is gm.

4. The unit of force is the dyne and its symbol is ^™^.

5. Force is measured by the product of the mass and the
acceleration, i.e., / = ma,

6. If a body that is free to move be affected by an unbalanced
constant, force, the motion will be uniformly accelerated.

7. At any given place, all freely-falling bodies have the same
acceleration. At sea level in the latitude of New York this
acceleration is 980 ^; therefore, since / = ma, a mass of 1 gm has
a weight of 1 X 980 = 980 dynes.

8. At any given place the weights of bodies are proportional
to their masses; therefore the masses of two bodies may be com-
pared by comparing their weights.

9. The density of a substance is its mass per unit volume.
Its symbol is ^,.

10. When masses are moved, or when their motions are changed,

MASS AND ENERGY 53

work is done; and the measure of the work done is the product of
the force and the corresponding displacement, i.e., W = fl,

11. The unit of work is the erg, and its symbol is ^^^^ .

12. Scientists are convinced that a perpetual motion machine
is impossible.

13. The doing of work implies the transfer of energy, and in
every such transfer two bodies are equally and oppositely affected.

14. The energy transferred is measured by the work done,
i.e., e = W,

15. The ratio of the useful work done to the total amount of
energy expended is the efficiency of the machine.

16. Energy is either potential or kinetic.

17. Kinetic energy may also be measured in terms of mass and

velocity, i.e., e = W= -r— .

18. Activity is the rate of doing work, and is measured by the
number of ergs done per second.

19. Engineering units are, for force, the pound-force; for work
or energy, the foot-pound; and for activity, the horse-power.

QUESTIONS

1. In what two ways may bodies differ in mass?

2. When different forces act on the same mass, what is the relation
of the forces to the corresponding accelerations? Illustrate by exam-
ples.

3. When the same acceleration is imparted to different bodies,
what is the relation between the forces and the corresponding masses?
Illustrate.

4. What kind of motion results from the action of a single or an
unbalanced constant force?

5. Define the cm-gm-sec unit of force and give its name and sym-
bol.

6. What is meant by the weight of a body?

7. What is the use of the equal arm balance, and why can it be
employed for this purpose?

8. Of what does work consist, and what is the numerical meas-
ure of an amount of work? Write the equation for work.

9. Name and define the cm-gm-sec unit of work. Give its symbol.
10. When is a body said to possess energy?

54 PHYSICS

11. What is meant by the term perpetual-motion machine? What
reason have we for believing that no such machine can be made?

12. What are some of the sources from which our supplies of en-
ergy ordinarily come?

13. When a body possesses energy, is all of it available for the
doing of useful work? Illustrate by some examples.

14. Define the terms kinetic energy and potential energy, and
give some examples of each kind.

15. What is the advantage of having an equation for kinetic energy
in terms of mass and velocity?

16. Explain the application of each of Newton's laws to the cases
of a rimner, a bicyclist, or an automobile.

17. If all the moving bodies on the earth, such as railroad trains,
steamships, and animals, were to travel eastward at the same time,
and continue to do so indefinitely, what would be the ultimate effect
upon the eastward velocity of the earth's rotation?

18. Define the engineer's units of force, work, energy, and activity.

19. What expression should be substituted for mass when engi-
neer's units are employed?

PROBLEMS

1. The masses of two loaded cars are 40,000 and 50,000 lb. re-
spectively; If a locomotive engine, exerting 2000 pounds-force on the
first car, gives it an average acceleration of 2.0 — j, what acceleration
would it give to the second car? What force would give the second
car the same acceleration as was given the first? Note: Friction is not
here considered. Each car would require a certain force to overcome
this, in addition to that required for the acceleration.

2. Five men, rowing a boat, give it in 30.0 sec a velocity of 15.0
— . What is the average acceleration? All otlier things remaining
the same, what velocity would be given the boat by three men? What
is the average acceleration in this case? What in each case is the dis-
tance traversed in the 30 sec?

3. A base ball has a mass of 140 gm, and is thrown from home
base to first, a distance of 2743 cm, in 0.90 sec. What is its velocity?
If the catcher applied the throwing force during 0.10 sec, what was
the average acceleration during that time? What was the force in
dynes? If it was stopped by the first baseman in 0.05 sec, what then
was the amount and sign of the acceleration? What force did it exert
on his hands?

4. What velocity and acceleration are given to a mass of 500.0 gm
by a force of 50,000 dynes applied for 10.00 sec? Tlirough what dis-
tance does the force act? How many ergs of work are done? How
many ergs of kinetic energy are stored in the moving mass?

MASS AND ENERGY 55

5. A base ball, mass 1^0 gm, was thrown vertically upward and is
caught by the thrower at the er i of 5.8 sec. Find the height to which
it rose, the velocity with which t, was thrown, its weight in dynes, the
work done on it, and the energy stored in it. If the force of the thrower
was applied during 0.05 sec, what was its amount, exclusive of that re-
quired to overcome the weight?

6. A block of marble, 1.00 X 0.50 X 3.00 m, has a mean density of
2.70 Q^. What is its mass? Express its weight in kilograms, and in
dynes. Express in ergs and in kilogram-meters the amount of work
that would be needed to lift it 5 m from the ground.

7. A brass cylinder has a mass of 122.50 gm, a diameter of 1.90 cm,
and a length of 5.10 cm. What is its density?

8. What is the volume of a copper ball whose mass is 130.0 gm, and
whose density is 8.87 gm?

9. A pound = 453.6 gm. How many dynes does its weight equal?
Find the weight of a 130 lb. boy in grams and in dynes.

10. To how many dynes is the weight of a kilogram equal? How
many ergs equal a kilogram-meter?

11. The pile driver. Fig. 20, has a mass of iron weighing 3500 lb.
This mass is raised by a steam hoisting engine to a height of 45 ft. and
dropped upon the head of a pile. Calculate the work in foot-pounds
required to raise the mass of iron to position. How much potential
energy has it when lifted, and how much kinetic energy when it strikes?
How much work does it do?

12. In the case of the pile driver, problem 11, calculate the time of
falling and the final velocity of the iron mass. From the weight and
velocity, calculate the energy when striking, and compare the result with
that calculated from the weight and the height. Which method of calcu-
lation for the energy would you choose if- both weight and height were
given, as in this problem? Which if the velocity were known, but not
the height?

13. How many pounds of water can be pumped per minute from a
mine 600 ft. deep by a 75 horse-power pump?

14. It is desired to raise ore from a mine 550 feet deep at the
rate of 3 tons per minute; what horse-power must the hoisting engine
be able to develop? How many foot-pounds of work would it do
per ton?

15. An automobile weighing with its load 2000 lb., starting from rest,
requires 22 seconds to attain a speed of 88 — , when it continues at uni-
form speed. Calculate its kinetic energy. What average horse-power
was used in putting it into motion? What other work had to be done?
How was the energy being expended after the speed became constant?

16. How many pounds of water must go over a fall each second in

56 PHYSICS

order to furnish 25 horse-power, if the fall is 10 ft. high and all the
power is to be used? How many cubic feet of water were used each
second if 1 cu. ft. weighs 62.5 lb.? What must be the cross-sectional
area of the stream at the fall, if the speed of the water there is 3 — ?

SUGGESTIONS TO STUDENTS

1. Consult the libraries on the life of Sir Isaac Newton, and prepare
a brief paper containing the facts that most interest you. This paper
may be read before the Physics class, published in the school maga-
zine, or offered as a theme in the English class.

2. Repeat Galileo's experiment, by throwing a block of wood and a
brick from a third story window.

3. How high can you throw a base ball? Take the time with a
stop watch, or with an ordinary watch, as accurately as you can; and
use equation (3). Plot a graph for the complete motion of the ball,
working out the distances and times for each of the seconds by equations
(2) and (3). Let the best throwers plot on the blackboard, to the same
scale, the graphs for their throws. Let the class compare and interpret
the changes of slope.

4. Get the necessary data by trial; and calculate your horse-power
(a) when going upstairs as fast as you can comfortably without a load;
(h) when carrying the greatest load that you can.

5. Devise a method of measuring on the wall of the house the greatest
height to which your lawn hose can throw water. Observe with a
watch the number of seconds taken by it to fill a gallon jar. Allowing
8 lb. to the gallon, calculate the number of pounds of water thrown
out in one sec. From this and the height, calculate the horse-power
that this stream of water could be made to furnish to a small water
motor.

CHAPTER III
COMPOSITION AND RESOLUTION OF MOTIONS

44. Up Grade. In Chapter II we have learned that a train
moves because the driving wheels of the engine are turned; and we
have studied the motions when the track is straight and level.
There still remain, however, many questions that need considera-
tion. Why must the engine work harder in ascending a grade?
How can we find the amount of this extra work? What is the
relation between the extra pull of the engine and the weight of the
train?

In all our previous study we have considered motions along
a straight line, i.e., in one direction or dimension only. The
questions just asked lead us to the consideration of what takes
place when a body has at the same time two or more different
motions. Since these motions may or may not be in the same
direction, the resultant motion may take place in two or three
dimensions, i.e., the path of the motion may be a plane curve, or
it may be twisted like the thread of a screw.

45. The Composition of Motions. One of the simplest cases
of two simultaneous motions of the same body is that of a man
walking lengthwise in a car that is moving uniformly on a straight,
level track. If the car is moving northward at the rate of 600 ^>
and the man walks in the same direction at the rate of 150 ^»
how far does the man travel northward in one second? In three
seconds? If the man faces about and walks southward in the
moving car at the rate of 150 ^, how far northward will he travel
in one second? In ten seconds?

From these examples it must be evident that in considering
motions we must take account of two characteristics of the motion,
namely, direction and magnitude. For this reason it is very
convenient to represent a motion by a straight line whose length

PHYSICS

and direction correspond to the direction and magnitude of
the motion. Thus, for the first case just considered, let ab, Fig.
22, represent the motion of the car northward: cd, which has the
same direction and is one-fourth as long, will then represent the
motion of the man with reference to a point in the car; and ad,
which is obtained by adding together ah and cdy
will represent the resultant motion of the man in
both direction and magnitude. 7

Similarly, in the second
case, if ab, Fig. 23, represent
the motion of the train, then
efy which has the opposite di-
rection and is one-fourth as
long, will represent the motion
of the man with reference to
a point in the car. Hence a/,
which is obtained by adding
together the two oppositely di-
rected lines, will represent the resultant motion of the man in
both direction and magnitude.

It is to be noted that in both cases this addition of the lines
is performed by drawing the first line with its proper direction and
magnitude, and then from the end of the first line drawing the second
with lis proper direction and magnitude. Then the line drawn
from the beginning of the first line to the end of the second represents
the resultant motion.

Fig. 22. Vectors

Fig. 23

46. Vectors. In order to indicate clearly the direction that
such a line represents, it is usually tipped with an arrow point as
in the figures. A line may be used in this manner not only to rep-
resent motions, but also to represent any sort of physical quantity
that has both direction and magnitude. A line that is used to
represent both the direction and magnitude of a physical quantity
is called a vector.

47. The Motions are at Bight Angles. Suppose now that
instead of walking northward or southward in the moving car,

COMPOSITION AND RESOLUTION OF MOTIONS

the man walks eastward across it. If the velocity of the car
is 600 ^ northward, and that of the man 150 ^ eastward, what
is the resultant motion during two seconds? Simple arithmetic
can not give us a solution that will determine the resultant both
in direction and magnitude. Therefore let us see what the graph-
ical method will do for us.

• In Fig. 24 let the distances northward be represented by the
ordinates, and the distances eastward by the abscissas, the scale
being 1 cm = 200 cm for each motion. Plot- y
ting the graph in accordance with the method
learned in Chapter I, we find that pi and 'p^ rep-
resent the positions of the man at the ends of
the first and second seconds respectively. Will
the points that represent his position at the end
of 0.5 sec and 1.5 sec also lie on the line O'p^,
Will this line include the points corresponding
to his position -at the end of 0.1, 0.3, 1.9 sec, etc.?
If we further subdivide the time unit and locate
points corresponding to any of the hundredths
of a second, will these lie on the line Opz? Is it
necessary to subdivide the time unit further in
order to show that the line O'p^ represents the
path of the man's motion as accurately as is
possible in the drawing?

Fig. 24 gives us the clew to an easy method of
finding the resultant of any two uniform motions;
for it is clear that the resultant is represented by
the concurrent diagonal Opj o^ the parallelogram Oy^ Ji^ ^2> whose
adjacent sides 0x2 ^^d Oz/j represent the two component motions
in both their directions and their magnitudes.

Fig. 24

Parallelogram op

Motions.

48. The Motions are not at BigHt Angles. Furthermore,
a little careful thought will make clear the fact that whatever may
he the angle between the component motions, the resultant is com-
pletely represented by the concurrent diagonal of the parallelogram
whose adjacent sides represent the two component motions both in
direction and magnitude. Thus, in Fig. 25, Ox represents one

PHYSICS

of two uniform motions, Oy the other, and the diagonal Op
the resultant. This construction is called the parallelogram

OF MOTIONS.

49. A Shorter Method. It may already have occurred to
the reader that the process of finding the resultant may be very
(/ _-yO much abbreviated. For it is evi-
dent that we can determine the
resultant Op2 (Fig- 24) just as defi-
nitely by means of the triangle OX2P2
as by the whole parallelogram. In
order to do this we have only to draw
the vector 0x2, representing the first
motion, and from its end Xj to draw
the vector a^jPa representing the sec-
ond motion; and then the line Op2>
which joins the beginning of the first vector with the end of the
second, is the vector that represents the resultant.

This method of construction is called the vector method. Fig.
26 is the diagram for a problem similar to that just considered.
Since the vector method is simpler than the parallelo-
gram method and is employed by physicists and engi-
neers, it will be used in the discussions that follow.
When we have found the vector that represents the
resultant, the actual magnitude of the resultant can
readily be found either from the diagram or by the
analytical method. --Thus, in Fig. 24 we can measure
the resultant vector Opj and we find its length to be
6.15 cm (nearly), and since in this case 1 cm of the
vector represents 200 cm traversed, the resultant dis-
tance is 6.15 X 200 = 1230 cm, the result by construc-
tion and measurement

Fig. 26

60. The Analytical Solution. To obtain the analyt-
ical solution we note that, since the two component motions are at
right angles to each other, the resultant is the hypothenuse of a right
triangle; and therefore, since the square of the hypothenuse is equal

COMPOSITION AND RESOLUTION OF MOTIONS 61

to the sum of the squares of the other two sides, the square of the re-
sultant is equal to the sum of the squares of the two components.
Hence, iii the example represented in Fig. 24, since one component is
1200 cm, and the other, at right angles to it, is 300 cm, the result-
ant = ^1200H 300* == 1236 cm, the result by the analytical
method. In general, i: R represent the resultant and A and B
the two components, then R = ^/Al^ + B\ provided that the com-
ponents are at right angles with each other.

Since the two numerical results just obtained represent the
same distance,, why are they not identical? Would the agreement
be closer if the diagram were constructed more carefully and on
a larger scale?

51. When the Angle between the Components is Oblique.
In this case the magnitude of the resultant can be obtained graph-
ically by the addition of vectors in the way just described. Having
drawn the resultant vector, we measure it in centimeters, and multi-
ply its length by the number of units that 1 cm represents on the
scale used in the diagram.

When the vector triangle is oblique, a purely analytical solution
is impossible without the use of the elements of trigonometry.
With a very little knowledge of trigonometry the solution is simple,
but those who have not this knowledge can always find the result-
ant by construction and measurement. In fact, it is generally
more convenient to get the resultant in this way; so that this
method of solution is very generally used by engineers.

52. Traveling Crane. The composition of three motions is
illustrated by a device used in shops where heavy castings or other
weights have to be lifted and carried from one position in the shop
to any other. This device, Plate II, is called a traveling crane
and consists of a steel bridge whose ends rest on little motor cars
which run on tracks supported by the side walls of the shop,
so that the crane can traverse the shop from one end to the other.
The bridge also carries another track along which another motor
car can run across the shop from one side to the other, while the
weight to be carried may be lifted to any desired height by means

62 PHYSICS

of a pulley hanging from the bottom of this car. The motor
cars and pulley are operated by electricity, steam, or compressed
air, and the operator controls them by levers sp that the car carry-
ing the pulley may be made to move either across or along the
shop while the weight is being raised or lowered by means of the
pulley. Thus the weight may move vertically while the car carries
it horizontally across the shop, or the crane may also at the same
time carry the weight horizontally along the shop. Therefore,
with this device it is possible to combine motions in three direc-
tions at right angles to each other.

53. Besolution of Motions. We have just seen how two com-
ponent motions may combine to make a single resultant motion.
In obtaining the solutions of engineering problems it is often con-
venient to conceive that an observed motion is the resultant of
two other motions that may have combined to produce it. Thus,
when by means of the traveling crane a casting is made to move
diagonally across the shop, it is clear that its actual motion is the
resultant of two> motions, one across and the other along the length
of the shop. In a similar way the motion of a railroad train up
a grade may be conceived as the resultant of two component motions,
one horizontal and the other vertical.

This separation of the actual motion into two conceived motions
leads us at once to the solution of several interesting problems
connected with the motion of bodies "up hill''; for let us suppose
that a train, running with uniform speed, is just beginning to
ascend a grade. How much more work must the engine do in
pulling the train up the grade than in pulling it for the same
distance and at the same speed along the level track? It is evident
that the amount of this extra work depends only on the steepness
of the grade. Suppose that the grade is 1 : 10, i.e., for every 100 cm
measured along the track, the track rises 10 cm. In order to
calculate the amount of extra work. done in pulling the train up
the grade, we shall, as stated, conceive the motion along the track
as the resultant of two component motions, one horizontal and
the other vertical. In Fig. 27, ac is the vector representing the
motion of the train while passing over 100 cm of track, and ab

Fig. 27. Resolution of Motions

COMPOSITION AND RESOLUTION OF MOTIONS OJ

and be are the vectors representing respectively the horizontal
and vertical motions of which we conceive ac to be the resultant.
Now, since ab is horizontal, no extra work is done by the engine
in imparting to the train
the motion represented
by that vector. But be
is vertical, and it is clear
that the engine can not
impart a vertical motion to the train without doing the work of lifting
the weight of the train. Hence the extra work done by the engine
in pulling the train up grade is the work done in imparting to the
train the motion represented by the vector be. But since the grade
is 1 : 10, the work done when the train traverses 100 cm of track
is that of lifting the train through a vertical height of 10 cm. Thus,
if the mass of the train is 2 X 10^ gna, then, since the acceleration
of gravity is 980 ^, we find by equation (4) that the weight of
the train is / = ma = 2 X 10* X 980 = 196 X 10* dynes. Since
the vertical displacement is Z = 10 cm, we have from equation
(5) for the work done JF = /Z = 196 X 10* X 10 = 196 X 10^'
ergs. This, then, is the extra work done on the train by the engine
for every 100 cn;i up grade along the track.

54. The Engine is Stalled. If the engine is not able to
supply the extra energy necessary to do this amount of work, it
will be stalled on the grade. Let us suppose that this has just
happened. What ten4ency to motion down grade has neutralized
that due to the engine pulling up grade? How does the magnitude
of this tendency depend upon the steepness of the grade?

55. Force Vectors. In these questions we are dealing with
forces, not motions; but forces have both direction and magnitude;
and therefore they can be represented by vectors, provided they
act at the same point. Thus, in Fig. 28 let the vector Om rep-
resent the weight of the train, which acts vertically downward.
This weight produces both a pressure against the track, and a tend-
ency to move downward along the incline. Hence to answer our
questions, we conceive the vector Om to be resolved into two
components, one perpendicular to the track and the other parallel

64 PHYSICS

to it. The first vector Op will then represent the pressure against
the track, in both direction and magnitude, and in like manner the
vector jmi will represent the tendency to move down the incline.
Now, since the pressure of the train on the track produces no

motion in the direction
of the vector Op, it must
be evident that this pres-
sure is balanced by an
equal and opposite pres-
sure. This equal oppos-
ing pressure will at once
be recognized as the reac-
tion of the track and earth. But if the component represented by the
vector pm were not balanced by an opposing force, the train would
move down the incline, i.e., in the direction of pm. At the instant
when the train is stalled, it is not moving either upward or down-
ward along the incline, and therefore what must be the direction
and magnitude of the opposing • force that prevents the down-
ward motion? How should the vector representing this oppos-
ing force be drawn in the figure?

Fig. 28.

Resolution op Forces on an
Inclined Plane

66. Balanced Forces. When two or more forces act simul-
taneously on a body in such a way that no motion results, these
forces are said to be in equilibrium. When forces
are in equilibrium the vectors that represent them
in the vector diagram, when added together, form
a CLOSED FIGURE. Thus, in the example just dis-
cussed there are three forces acting, namely, the
weight of the train, represented by Ow (Fig. 28), the
resistance of the track, represented by pO, and the
pull of the engine, represented by mj). If these three
vectors be added by laying off one from the end of an-
other, each with its proper direction and magnitude, and
in any order, they form a closed triangle as in Fig. 29. forces in

On the other hand, and in general, if we have any ' quilibrium
number of forces not in equilibrium acting on a body, and wish

TO FIND THE FORCE THAT WILL HOLD THE SYSTEM IN EQUILIBRIUM,

COMPOSITION AND RESOLUTION OF MOTIONS 65

we add successively the vectors that represent the given forces,
and draw a line from the end of the last vector to the beginning of the
first. This line will then be the vector that repre-
sents the force sought, in both direction and magni-
tude. Thus, in our example, if we know the weight
of the train and the resistance of the track, and if
we wish to find by this method the pull of the engine
which will hold the train stationary on the grade,
we add together the vectors rs and st which repre-
sent the two known forces as in Fig. 30; then
the line tr is the vector sought.

This method of finding the force that is able
to hold a system of other forces in equilibrium is
very useful in engineering practice in connection with the design of
bridges, roof trusses, and other structural work in which it is nec-
essary to determine how strong a beam or tie must be in order to
resist the given stresses and hold them in equilibrium.

57. The Pull of the Engine and the tendency down the
incline are in equilibrium; therefore we can determine the magni-
tude of either of them, either graphically or analytically. Thus,
since the mass of the train is 2 X 10* gm, the vector Om (Fig. 28),
2 cm long, represents the weight, namely, 196 X 10' dynes; and
hence 1 cm in the diagram represents 98 X 10' dynes. The length
of the vector pm is found by measurement to be 0.2 cm, and
hence it represents a force of 0.2 X 98 X 10' = 196 X 10* dynes,
which is the magnitude sought.

68. To Get the Analytical Solution we must notice that the
triangles .450 and Omp (Fig. 28) are similar. (Why?) Therefore

^^AC' (^^^^ • ) ^"* ^^^^^ *^^ ^^^^^ ^^ AC ^ To ^^'^yP^^^"

esis, it follows that ^Ic— = — . (Why?) Whence pm = ^V Om..

Since Om = 196 X 10\ pm = yV X 196 X 10' = 196 X 10' dynes,
as in the preceding paragraph.

^^ PHYSICS

69. Less Force: Greater Distance. Now, we have learned
in Art. 53 that the extra work done in pulling the train 100 cm
along the incline is the work done in lifting the train through a
vertical height of 10 cm, i.e., in the case there considered, it is
W = Jl= 196 X 10* X 10 = 196 X 10'" ergs. But we have just seen
that the pull' of the engine is 196 X 10* dynes; and therefore, when
this pull is exerted through a distance of 100 cm, the work done is
W = /7'= 196 X 10* X 100 = 196 X 10'* ergs, as it should be.
It will be noted, however, that although the amount of work is
the same as that previously calculated from the vertical lifting of
the train, the force of the engine is only -^^ of that which would be
required to lift the train vertically through the 10 cm. The advan-
tage of using AN INCLINED PLANE is therefore apparent, since we
see that by means of it we can do a given amount of work with a
smaller force than would be required without it. Hence such an
inclined plane is said to furnish a mechanical advantage.
This mechanical advantage is defined as the ratio of the resistance
overcome to the effort applied. In the case of the inclined plane,
when the effort is applied parallel to the length of the plane, the
measure of the mechanical advantage has been shown to be the
ratio of the length to the height.

Thus, in general, if h represent the height of the plane, I its
length, R the vertical resistance to be overcome, and / the force
exerted parallel to the plane (c/. Art. 58 and Fig. 28), then

? = i

/ ~ h'

This is the analytical expression for the mechanical advantage of
the inclined plane when the effort is applied parallel to its length. It
may also be written Rh = fl, which expresses analytically the fact
that the amount of work done by the force applied parallel to the
plane is the same as that which would be done if the body were lifted
vertically through a distance equal to the height of the plane.

It has probably occurred to the reader to ask, Since the engine

. is stalled part way up the grade because its pull is no greater than

the pull of the train dow^l grade, why does the train ascend the grade

at all? The answer is that when the train reached the grade it

COMPOSITION AND RESOLUTION OF MOTIONS

was moving with a uniform velocity; hence it had kinetic energy
whose amount is determined by equation (6) as e — im F*. It
was this kinetic. energy that did the work of lifting the train; and
when this energy was expended, the unaided force of the engine
could carry the train no farther.

60, Definitions. Some of the ideas considered in this chapter
occur so frequently that » we shall do well to frame definitions
for them.

The single motion that will produce the same effect as that
produced by two or more
motions is called a re-
sultant MOTION.

The several motions
that combine to produce
the resultant are called

COMPONENT MOTIONS.

The process of find-
ing the resultant of two
or more motions is called

the COMPOSITION OF MO-
TIONS.

The process of find-
ing the components when the resultant is known is called the

RESOLUTION OF MOTIONS.

By substituting the word force wherever the word motion is
used, we can frame a similar set of definitions for the composition
and resolution of forces.

The single force that will hold two or more others in equilib-
rium is called their equilibrant. The equilibrant of any set of
forces is equal in magnitude to their resultant, and opposite in
direction. The point of application of the resultant is identical
with that of the equilibrant,

61. The Problem of the Besolution of a Motion, or of a force
acting at a given point, into two components is indeterminate
unless something more than the resultant is given. Stated

Fig. 31. Inclined Railroad, Pike's Peak

68 PHYSICS

geometrically, the problem is : given one side of a triangle, to find
the other two. Evidently, we can construct any number of triangles
that will satisfy this condition.

A little attention to the geometry of the triangle shows that
in addition to the direction and magnitude of the resultant, we
must know of the components either (1) both magnitudes (three
sides) ; or (2) both directions (a side and two adjacent angles) ;
or (3) one magnitude and one direction (two sides and an angle).

SUMMARY

1. Any linear motion may be represented in both direction
and magnitude by a straight line called a vector.

2. The vector of a resultant motion is found by adding the
vectors of the component motions.

3. If two component motions are at right angles to each other,
the resultant motion is numerically equal to the square root of
the sum of the squares of the two component motions.

4. Any motion may be resolved into two or more component
motions.

5. In order to resolve a motion into two components, we must
know of the components either (1) both directions; or (2) both mag-
nitudes; or (3) one direction and one magnitude.

6. The mechanical advantage of an inclined plane is equal
to the length of the plane divided by its vertical height.

7. The work done in moving a body up an inclined plane is
equal to the work done in lifting the same body vertically through
a distance equal to the height of the plane.

8. Forces that act at a given point may be represented by
vectors.

9. When the vectors that represent any set of forces in equi-
librium are added together in any order, they form a closed polygon.

10. The vector that represents the resultant of a number of
forces not in equilibrium is found by adding in any order the vectors
of these forces, and drawing a straight line from the beginning of
the first vector to the end of the last.

11. The equilibrant of any set of unbalanced forces is equal
to their resultant in magnitude, but opposite in direction.

COMPOSITION AND RESOLUTION OF MOTIONS 69

QUESTIONS

1. Explain what a vector is, and how it may represent completely
any physical quantity that has direction and magnitude.

2. Explain how vectors may be added in order to find the resultant
of two motions when these two components have: 1, the same direc-
tion; 2, opposite directions; 3, directions that are neither the same nor
opposite. How is the magnitude of the resultant motion found after
the resultant vector has been drawn?

3. Explain the manner in which the analytical expression for the
resultant of two motions may be found when the components have
directions at right angles to each other.

4. Describe a traveling crane, and explain how with it a body
may be given two or three different motions at the same time.

5. With the aid of a vector diagram, explain how the motion of a
body up or down an inclined plane may be conceived as made up of two
components, one horizontal and the other vertical.

6. When the weight of a body and the vertical height of an in-
clined plane along which it is to be lifted are known, what is the amount
of work done in lifting it along the plane?

7. How does it follow from the vector diagram in question 5 that
the work done in lifting the body up the incline is equal numerically
to the weight of the body multiplied by the vertical distance through
which it is lifted?

8. Show by a vector diagram that the weight of a body is to the
force necessary to hold it in equilibriima on an inclined plane as the
length of the plane is to the height.

9. Write an equation which expresses this relation. How does
this equation show that the inclined plane furnishes a mechanical ad-
vantage?

10. By means of this equation, show that the work done by a force
pushing the body upward along the incline is equal to the work that
would be done if the body were lifted vertically through a distance
equal to the height of the plane.

11. Show how the vector for the resultant of any set of forces
acting at a point may be found.

PROBLEMS

1. A man rows a boat with a velocity of 200 ^"^ southward in a

"^ sec

stream that has a velocity of 100 — southward. Find the resultant

velocity of the boat by the vector method, and also by calculation.

2. Find the resultant velocity by both methods when the boat is
rowed northward, the speeds remaining the same.

70 PHYSICS

3. Find the resultant velocity by both methods when the man keeps
the boat headed due westward, and does not try to resist the current,
but rows with the same speed as before.

4. A boy rows a boat with a velocity of 3 ^^^^, keeping it headed
across the stream, and not attempting to resist the current. The ve-
locity of the current is 4 E^. Find the resultant velocity of the boat
by both methods.

5. Suppose that the width of the stream in problem 4 is }
mile, how many minutes will it take to cross the stream? How far will
the boat drift down stream? How far will it actually travel along the
resultant path?

6. The boy wishes to cross the stream in the same time as in prob-
lem 5, but intends to land at a point directly opposite the starting
point. Show by vectors the direction in which he must keep the boat
headed. By both methods find the speed with which he must row in
order that the boat may move in a straight line from the starting
point to the landing point. How far up stream would his row have
taken him if there were no current?

7. If the traveling crane, Plate II, carries the pair of wheels across
the shop at the rate of 1.2 — , while it moves along the shop at the

rate of 1.6 — » ^^ ^^® resultant velocity by both methods,
sec

8. Suppose that in addition to the other two motions of problem 7

the crane pulley rises vertically at the rate of 0.5 — , what is the final
resultant velocity of the pair of engine wheels?

9. A trolley car weighs 10 tons and moves 1000 ft. along a grade
that rises 1 ft. in every 100; how much work must the motor do? What
is the mechanical advantage of the plane? What is the amount of the
force that moves the car up the grade?

10. If in problem 9, the speed was 50 — , what was the horse-power?

11. The height of an inclined plane is 2 m and its length 10 m; the
weight of a barrel that is rolled up this plane is 150 Kg; required the
mechanical advantage of the plane, the number of kilograms-force
exerted, and the number of kilogram-meters of work done.

12. A ball rolls down a smooth inclined plane whose length is 10^

cm and whose height is 10* cm. If the ball had fallen vertically, its

acceleration would have been 980 -^. Conceive this acceleration to

sec2

be made up of two components, one along the plane, and the other
perpendicular to it. Determine both graphically and by calculation
the acceleration of the ball down the incline.

13. The weight of a kite is 2X10^ dynes; the pull on the string is
4X10^ dynes and makes an angle of 60° with the vertical. Find the
resultant pull on the kite. What must be the direction and magnitude
of the force that keeps the kite in equilibrium?

COMPOSITION AND RESOLUTION OF MOTIONS

14. In Fig. 32, ab represents the direction of the keel of a boat,
and the line d the direction of the sail, and / is a vector that repre-
sents the effective pressure of the wind, 200 kil-
P ograms-force. By the vector _

method, find the force that
urges the boat forward, and also
that which urges it sideways.
How is sideways motion pre-
vented?

15. By the vector method,
find the nimiber of kilograms-
force with which the beam or
strut ab, Fig. 33, must push,
and that with which the tie rod cd must pull in order to keep the 40
Kg ball in equilibrium.

Fig. 33

16. A ball is thrown upward with a velocity of 4900

For

how many seconds will it rise before its velocity is reduced to zero by
the negative acceleration of 980 ^j? What is the distance to which
it rises in this time? How long will it take to reach the starting point?
Calculate the velocity at the instant of reaching the starting point and
compare this with the velocity with which it is thrown.

17. Calculate the distances traversed by the ball of problem 12
at the ends of the successive seconds, and plot the graph for its motion.
Describe the changes of slope. What is the slope at the maximum
distance, or highest point? Compare this graph with the path described
by a body thrown obliquely upward.

SUGGESTIONS TO STUDENTS

1. Point your lawn hose at an angle of 45° elevation; note the

path of the drops of water. Assuming 1000 — as the initial velocity

of the water, find its vertical and horizontal components. Assume that

the horizontal velocity is uniform and that the vertical velocity has a

negative acceleration of 980 ~. Calculate the distances traversed ver-
sec2

tically and horizontally at the end of each fifth of a second. Plot a
graph with the vertical distances for ordinates, and the horizontal
distances for abscissas. Is this graph the same sort of curve as the
actual path of the water? Are you justified in inferring from the com-
parison that the vertical velocity was uniformly accelerated and the
horizontal velocity uniform in the case of the water jet?

2. Bring a toy sail-boat to the class room to illustrate problem 14.
With the aid of vectors, can you find an explanation of how such a bo»<^
can "beat against th^ wind"?

72 PHYSICS

3. Bring in sketches or photographs which show struts and ties used
in ways similar to that mentioned in problem 15. You will find them
on electric light poles, supporting signs, in the frames under cars, in
roofs, in bridge trusses, in jib cranes, in locomotive cranes, in bicycle
frames, etc. Try to draw the vector diagrams for each case brought in.

4. How high can you throw a ball? Note with a watch the total
time taken by the ball in rising and falling. Also calculate the initial
velocities (c/. problems 16 and 17). Place on the blackboard the names
of the best throwers, with velocities and distances attained.

CHAPTER IV

MOMENTS

62. How Botation is Caused. Thus far^ we have consid-
ered motion of translation only. We are now ready to take up some
of the conditions under which rotary motion may occur; and the
railroad train furnishes us with several questions whose answers
will help us to describe accurately some relations about which we
already have some general ideas. How is the translatory motion of
the piston converted into rotary motion of the drivers? And why
are the drivers of the fast passenger engine made large, while those
of the freight engine are made small?

In order to find the answers to these questions, let us consider
the diagram. Fig. 34. When the connecting rod pushes on the

Fig. 34

crank pin at nj* or pulls at n^, it is evident that it can not cause the
wheel to rotate, but produces only a useless strain on the moving
parts. When, however, the crank pin is anywhere above or below
the line nn^, the pull or push of the connecting rod will cause the
wheel to revolve. Furthermore, common expe;rience tells us that
the force of the connecting rod is more and more effective as the
distance from the center of the wheel to that rod increases. There
is, then, some relation between the effectiveness of a force in pro-

PHYSICS

Fig. 35. The Moments are Balanced

ducing rotation, and the distance from the axis of rotation to the
line of direction in which the force acts. How shall we measure
the effectiveness of a force for producing rotation?

Let us suppose that the board in Fig. 35 is supported at the
middle. It will then balance, so that its weight may be left out of

the problem. Sup-
pose that a boy,
whose weight is 20
kilograms, sits 100
cm from the axis.
If a girl is seated
100 cm from the axis
on the other side, the
boy's weight can just
hold the girFs in
equilibrium, provid-
ed her weight is also
20 kilograms. Now, if the boy's weight is 25 Kg and his distance
from the axis is 100 cm, he can balance another at 100 cm whose
weight is 25 Kg; and so on. Thus in general it appears that the
effectiveness of a force at a constant distance from the axis of rota-
tion, is directly pro-
portional to the mag-
nitude of the force.
Again, suppose
that a boy's weight
is 20 Kg, and that
he is distant 200 cm
from the axis. He
can now balance two
children at 100 cm,
each having the
weight of 20 Kg
(Fig. 36). If the 20 Kg boy is distant 300 cm from the axis, his
weight will be as effective in turning the board as is a weight of
60 Kg at 100 cm; and so on. Thus, in general, if the distance from
the axis varies, while the force remains constant, the effectiveness

Fig. 36. Moment Equals Force X Arm

MOMENTS

of the force in producing rotation about that axis is directly pro-
portional to the ARM OF THE FORCE, i.e., to the perpendicular dis-
tance between the axis and the line of direction of the force.

63, Moment of Force. The effectiveness of a force in pro-
ducing rotation about an axis is called the moment of the force
about that axis.

Since we have seen that the moment of a force is directly propor-
tional to the magnitude of the force when the arm is constant, and
directly proportional to the arm when the force is constant, it is
clear that the appropriate numerical ineoMire of the moment of a
force is the product of the force and its arm with respect to the given
axis.

Returning to the case of the locomotive drivers (Fig. 34), we see
that the measure of the turning effect is the force F^, applied to
the crank pin, multiplied by the perpendicular distance from
the center of the wheel to the middle line or axis nn^ of the connect-
ing rod.

Since now we know how to calculate the moment of a force, we
shall be able to consider a few problems that will lead us to the
statement of some very important principles, and will also enable
us to answer the questions that were raised concerning the relative
sizes of driving wheels for passenger and freight engines.

64. The Lever. Suppose that the man in Fig. 37 is to do
the work of lifting, Avith the lever, a stone which weighs 100
Kg. He pushes vertically
downward at one end with
a force which we will call /.
The fulcrum, i.e., the axis
p (Fig. 38) about which
the lever turns, is distant
40 cm from the center of
the stone and 200 cm from
the man's hands. The
moment of / with respect ^^°- '^- ^''^ ^^^«

to the fulcrum is / X 200, and that of the stone's weight is 100 X 40.
If the moment of / is just sufficient to keep that of the stone's

PHYSICS

weight in equilibrium, then / X 200 = 100 X 40. Whence,
finally, / = 20 kilograms-force = 20 X 1000 X 980 = 196 X lO'
dynes.

The force that will move the stone must, of course, be somewhat
greater than this, because some unbalanced force is required to
produce the acceleration.

The equation may be written:

100 200 5
/ " 40 "^ 1 '
which states that the mechanical advantage of this lever is 5 (cf.
Art. 59).

65. The Work Done by the Lever is easily calculated. When
the lev^r is moved. Fig. 38, the point s describes an arc with a radius

of 40 cm, and moves, say, from s
to s', while the point h describes
a similar arc with a radius of 200
cm, going from A to A'. Suppose
the vertical distance sm through
which the stone is lifted is 10 cm.
The effort, at the same time, acts
through the vertical distance hn.
If the stone weighs 100 Kg, or
98 X 10" dynes, calculate how
many ergs of work are done in lifting it through 10 cm.

Now, since the right triangles msp and nhp are similar (Why?),

— == -TTT = T- Since sm = 10 cm, what is the value of
5m 40 1

An? Thus it appears that, although by means of this lever we are
able to dok the work of lifting a stone with a force that is only one-
fifth of the weight of the stone, this force must be exerted through a
distance or displacement five times as great as that through which
the resistance is moved.

The work done by / is / multiplied by its displacement, or
(196 X 10^) X 50 = 98 X 10^ ergs. How does this amount of work,
done by the man, compare with that done on the stone as previously
calculated?

40 jn ./

Fig. 38. The Lever Diagram

MOMENTS 77

A lever is often used in another way. as in Fig. 39, when the
fulcrum is at one end, and the resistance between, — the effort being
applied at the other end as before.
In this case the application of
the principle is entirely similar;
but the possible mechanical ad-
vantage is greater, because the /
lever arm of the effort is longer, t
The moment of the effort with
respect to the fulcrum is now
/ X 240, and that of the resist-
ance is 100 X 40 as before, the

... - ^ . .1 Fio. 39. Another Lever Diagram

mechanical advantage is there-
fore found from the equation / X 240 = 100 X 40. Whence

40
/ = 100 X oTrj = 16.66 Kg-force. The mechanical advantage

240
in this case, therefore, is — , or 6. The geometrical construc-
tion by which the number of ergs of work are found and proved
equal is much like the preceding, except that the similar trian-
gles are differently placed. It is easily seen from the figure that

-f— — -777 = T> and that the effort X 60 = the resistance X 10.
sm 40 1

66. The Lever Principle. In the examples just worked out,
we have learned four things about the lever. Other problems
involving levers can be solved in a similar manner. The four
things that we have learned are:

1. The lever is in equilibrium when the moment tending to
turn it in one direction is equal to that tending to turn it in the opposite
direction.

2. The mechanical advantage of a lever is equal to the effort
arm divided by the resistance arm.

3. The mechanical advantage of the lever may also be obtained
by dividing the displacement of the effort by the displacement of the
resistance.

4. The work done by the effort is equal to tlie work done on the
resistance.

78 PHYSICS

These statements may all be verified by very simple experiments
in which the forces and distances are measured when various
kinds of levers are in equilibrium. In such experiments and
problems, it must be noted that whenever the weight of the lever
itself is at all comparable in magnitude with the other forces
involved, it also must enter into the calculation.

67. Equilibrium of Parallel Forces. Another impoiiant fact
concerning the lever (Fig. 38) is sufficiently obvious without
argument. The two downward forces must produce a downward
pressure on the fulcrum; this downward pressure is their resultant,
and is equal in magnitude to their sum. Hence it is evident that the
fulcrum must exert an upward resistance which is the equilibrant
of this resultant, and which is therefore equal in magnitude to the
sum of the downward forces. It is also clear from Art. 65 that
the point of application of this equilibrant divides the line joining
the points of application of the two forces into segments that are
inversely proportional to the magnitudes of the forces. There-
fore when a system of parallel forces in one plane acts on a body,
the condition that must be fulfilled in order that no translatory
motion may take place is that the sum of the forces acting in
one direction be equal to the sum of those acting in the opposite
direction.

Similarly, the condition that must be fulfilled in order that
no rotation may take place is that there be no resultant nioment,
i.e., that the sum of the moments tending to turn the system in one
direction about any point be equal to the sum of the moments
tending to turn it in the opposite direction about the same point.

It will easily be understood that these conditions for equilib-
rium which we have seen apply in the case of three parallel forces,
must hold for any number of such forces; for clearly if there is no
unbalanced force, there can be no translation; and if there is no
unbalanced moment, there can be no rotation.

68. Illustration by a Problem. If we wish to determine the
single force that will hold a system of known parallel forces in
equilibrium, we can do so with the help of these principles. For

-1Q.QL — »50.

MOMENTS 79

example, suppose it is required to lift the shaft with its pulleys.
Fig. 40, by applying a single vertical force in such a way that the
shaft will remain in a horizontal position as it rises. How great
a force will be neces-
sary, and at what
point must it be ap-
plied? The indi- .^_ M^_^_

cated weights of the

wheels and the shaft A

are the known forces, T \ ^o

and their respective So

*■ ■ Fig. 40. The Shaft Remains Level

distances from A^

the end of the shaft, are the known arms. We may assume that the
bar is uniform, sd that its weight acts at its middle point, as shown
in the figure. The lifting force / and its arm r are to be deter-
mined. From the dimensions on the diagram we see that the sum
of the downward forces is 50 + 60 + 30 + 80 = 220. The
required upward force, therefore, must be equal to this sum, or
/ = 220 Kg-force.

Since the condition for no rotation is that the moments, taken
with respect to any point, be balanced, we may select the left end
of the shaft as the most convenient point of reference. The sum
of the moments with respect to this point is, then, evidently
(50 X 25) + (60 X 200) + (30 X 250) + (80 X 350) = 48750.
This moment must be counterbalanced by that of the upward force
of 220 having the unknown arm r. Hence, (220 X r) = 48750.
Whence r = 221.6 cm, i.e., the force necessary to hold the shaft
in equilibrium is 220 Kg-force; and it must be applied at a point
221.6 cm from the left end of the shaft. Of course some addi-
tional force will be required to produce the acceleration when the
shaft is moved.

69. The Equilibrant of Any Number of Parallel Forces may
be determined in a manner similar to that used in the example
just given. Since we can form two equations in which all the
forces appear, we may determine either one force and one arm, as
in the example, or two forces whose arms are known.

PHYSICS

Fig. 41. A Fast Engine

70. The Locomotive Drivers. Let us apply the principles of
the lever to the driving wheels of the locomotive. Let F^ repre-
sent the pull of the connecting rod nn^ (Fig. 34); and let r^, which

is the perpendicular dis-
tance from the center
of the wheel to nn^, rep-
resent the arm of this
force. Also let Fj rep-
resent the push mFj,
exerted by the rim of
the wheel along the
track; and let rg, the
radius of the wheel,
which is the perpendicular distance from its center to mF2, repre-
sent the arm of the push Fj. Then from the equation for the

F r .

mechanical advantage of the lever, -^r = — • This equation shows

^ t ^
that the horizontal push at the rim of • the driver is less than that
exerted on the crank pin, in the same proportion as the distance
T^ of the crank pin from the center is less than the radius ra
of the wheel.

Now, an engine with large driving wheels can develop greater
speed than can one with smaller ones, because the circumferences
of the drivers are large; and the engine will go farther, for each
stroke of the piston. But in this case our equation shows us that,
other things being equal, the push that can be exerted on the
track is proportionately
less; because rj, the
radius of the driving
wheel, is increased in
the same proportion as
is the circumference.

On the other hand,
an engine which is to
haul a long and massive freight train, must be able to exert a very
great horizontal push on the track, and therefore rg must be made
smaller in proportion to r^. This necessitates smaller driving

Fig. 42. A Powerful Engine

MOMENTS

wheels, giving less speed. Also, since the aim is to get as much
power as possible, the engine must not only have large and power-
ful steam cylinders, but must also be very heavy, so as to exert
sufficient pressure on the track; otherwise the driving wheels will
slip, and the engine will not be able to move the train.

71. Weight and Center of Mass. Some very important
applications of the principles pertaining to parallel forces are found
in the action of gravity on bodies.
For gravity tends to pull each
particle of a body toward the
center of the earth; therefore,
the gravity forces that act on all
the particles of a body are prac-
tically parallel, and their result-
ant is the weight of the body.

Now, when any body is acted
on by a system of forces affecting
all its particles, and all in the
same direction, there is a point
so situated that the moments of
all those forces will be balanced
about any axis that passes
through this point. This point,
which is evidently the point of
application of the resultant of all the parallel forces acting on the
particles of the body, is called the center of mass. The center
of mass of a body is, therefore, the point of application of its
weight, and hence it is often called the center of gravity.

72. Equilibrium. // a force acting vertically upward, and equal
to the weight of a body, he applied so that its line of direction passes
through the center of mass, the body will he in equilibrium under
the action of this force and its weight

Thus suppose that c (Fig. 44) is the center of mass of a body sus-
pended at a point s, about which it is free to rotate, as is the case
with swing (Fig. 43). Then if the body has been slightly displaced

FiQ. 43. The Swing

PHYSICS

from the position wherein c is vertically below s, there is a moment
which is equal to the product of its weight w and the distance sb,
and which will return it to that position. In what position will
such a suspended body be in equilib-
rium? The vase and the pitcher, Fig.
45, are in equilibrium; what moment
tends to return each of them, when it
is slightly tilted?

73. The Stability of a Body like the
vase or the pitcher, which rests on a
BASE, is measured by the amount of
work that must be done in overturning
it. A little consideration will show that
the amount of this work may be de-
termined as follows: With o as a cen-
ter and a radius equal to ac, describe an
arc. This arc is the path that the cen-
ter of mass c will describe when the
body is overturned about the point or
axis represented by a. When the cen-
ter of mass c is in the vertical line that
passes through a, the body will be in unstable equilibrium, and
the smallest further displacement will overturn it. From c draw
a horizontal line intersecting oc' at a point 6. Then &(/ is the ver-
tical distance through which the center of mass must be raised in
order to overturn the body; and the work done is found by mul-
tiplying the. weight by this vertical distance.

Other things being equal, if the base of the body were smaller,
or if the center of mass were higher, as in the case of the vase,
what would be the effect on fee', and on the work done in over-
turning the body?

Answers to questions like these lead to the general conclusion
that, other things being eqical, the larger the base of a body, and the
lower its center of mass, the greater is its stability.

Fig. 46 represents a sphere of uniform density, whose center
of figure is therefore its center of mass. Show that it is in equi-

FiG. 44.

The Swing Dia'

GRAM

MOMENTS

Hbrium in any position on a level plane. Show also that when the
plane upon which it rests is slightly tilted, there is a component

Fig. 45. Stability is Measured by Work

of force urging it down the plane, and also a moment of force tend-
ing to rotate it.

74. Determination of Center of Mass. The foregoing prin-
ciples of equilibrium enable us to find the center of mass of a body
by experiment; for if the body be
freely suspended from a point near
one of its extremities, it will come to
rest in the position wherein the arm of
its weight becomes zero (cf. Fig. 44).
This position is evidently that in
which the center of mass is in the
vertical line passing through the
point of support. If this line be
indicated by a plumb line, and
marked on the body, we know that it contains the center of
mass.

Fig. 46. The Ball Has No Sta-
bility

PHYSICS

If, now, another point of suspension be selected, and a new
vertical line marked in the same way, it must be apparent that the
center of mass, since it is in both these lines, can be nowhere else
than at their intersection.

Another way of finding the center of mass of a flat, thin body,
such as a piece of pasteboard, is to balance it flatwise upon a straight-
edge, and mark on it the axis upon which it balances. This axis,
in accordance with the definition, must contain the center of mass.
Therefore, if another axis about which the moments balance be
located in the same way, the center of mass
is the* point in which the two axes intersect,
for it will be found that every other axis on
which the body will balance passes through
this point.

These experiments are of great con-
venience in connection with certain engi-
neering problems; for it is often neces-
sary to find the center of mass of a part
of a machine, or of a piece of some struc-
tural work, in order that it may be de-
signed so as to be in equilibrium under
the given conditions.

For example, there must be placed
on a locomotive driver (Fig. 47) a coun-
terpoise having a moment exactly equal to that due to the con-
necting rod or side rod used in turning the wheel. This is be-
cause if the moments of all the rapidly rotating parts are not thus
accurately balanced against each other, the system will wabble and
produce a wasteful and even destructive strain on its axis of rota-
tion. In order to place the counterpoise properly, its center of
mass must be known ; and since it is not a regular body, this deter-
mination can not easily be made by geometry. The usual practice,
therefore, is to cut out a pasteboard model to a certain scale,
and experimentally determine the center of mass of this model.
The center of mass of the real object is then easily located, for it
is the point of the real object that corresponds to the center of mass
of the model.

Fio. 47. The Driver Has
A Counterpoise

MOMENTS 85

The same method is used in order to get the position of the
center of mass of half of a stone arch, so that the moment due to
its weight can be calculated.

75. Mechanical Advantage of a Composite Machine.
Before leaving the study of the- applications of the lever principle,
let us consider how we can find the mechanical advantage of a
contrivance like that in Fig. 48, in which the lever principle and
that of the inclined plane are used simultaneously.

In pulling the safe up the inclined plane whose height is 100 cm
and whose length is 400 cm, the mechanical advantage obtained

by means of the plane (c/. Art. 59) is t- = -jt^ = 4; i.e., the

weight of the body that can be moved along the incline is four
times the pull of the rope. Fur-
thermore, the effort, which is ap-
plied to the end of the crank,
has a greater lever arm than has
the pull of the rope. The effort
arm in this case is the length of
the crank, and the resistance ann
is the radius of the axle. There-
fore we obtain by this device a /'«• f- Composite Machine
. •' /PA ^ windlass and an inclined plane,
mechanical advantage (c/. Art.

65), which is equal to the ratio of the length of the crank to the
radius of the axle. If these two lengths are 50 cm and 10 cm

/ 50 .

respectively, then -^ = — = 5, i.e., the pull on the rope is five

times as great as the effort applied at the crank handle.

Now, if the man applies to the crank handle a force equal to
the weight of 40 Kg, it is clear that since the mechanical advantage
of the windlass is 5, the pull on the rope is equal to 40 X 5 kilograms-
force. Furthermore, since the mechanical advantage of the
inclined plane is 4, the weight that can be lifted along the
incline by this pull of 40 X 5 is, (40 X 5) X 4 = 40 X 20 = 800
kilograms-weight. Thus we see that the mechanical advantage

of the combination is ;= — ; — = -77- = 20. This mechanical

\ effort 40

86 PHYSICS

advantage of the combination can be quickly obtained by multiply-
ing together the mechanical advantages of the elementary parts;
e,g,, 5 X 4 = 20.

Similar reasoning applied to other problems shows that this
method of procedure will give the mechanical advantage of any
composite machine, no matter how complicated it may be. Hence
in general we can find the mechanical advantage of any composite
rrmchine by multiplying together the mechanical advantages of the
several elementary machines of which it is composed.

76. The Law of Machines. We have seen that for the
inclined plane and for the lever, the work done by the effort
is equal to the work done on the resistance. Let us now
see if this is true for the combination of these two devices.
In the case just discussed, the effort was supposed to be
40 kilograms-force = 40 X 1000 X 980 = 392 X 10^ dynes. The
resistance was 800 kilograms-weight = 800 X 1000 X 980 = 784 X 10^
dynes. The distance Ij^ through which the effort acts in one
revolution of the handle is Z^ = 27r X 50 cm; and the vertical
distance ^ through which the weight is lifted may be found as
follows : For one turn of the crank, the rope is drawn up a dis-
tance equal to the circumference of the axle, or 27r X 10 cm.
Evidently the safe moves the same distance up the incline. But
since the height of the plane is one-fourth of its length, the
vertical distance through which the safe is lifted is one-fourth
of the corresponding distance that it moves along the incline;

, 27r X 10 cm . ^ _ _
I.e., I2 = T = 27r X 2.5 cm.

The work done by the effort, therefore, is found to .be
/i^i = (392 X 10^) X (27r X 50) = 392 X 10^ X tt. That done on
the resistance is f^k = (784 X 1(f) X (27r X 2.5) = 392 X 10^ X tt.
Thus the two amounts of work are equal.

Similar reasoning proves that this principle, which we have
demonstrated in the cases of an inclined plane, of a lever, and of a
combination of the two, applies to all machines whatsoever. It
is usually called the law of machines, and is stated as follows:
The product of the effort and the distance through which it acts is

MOMENTS 87

equal to the jyroduct of the resistance and the distance through which
it is overcome; or, the work done by the effort equals that done on the
resistance. In symbols,

fA-hh- (7)

It should be noted that the distances l^ and l^ mu^st always be
measured in the directions of the corresponding forces. Also, in
applying this statement to any particular case, it should be bome
in mind that the total work done invariably includes some useless
work against such resistances as friction, rigidity of parts, inertia^
reaction, and resistance of the air; so that in order to make the state-
ment precise and perfectly general, this useless work must be under-
stood to be added in with the useful work. When this has been
done, it is invariably found that the work done is the exact equiv-
alent of the energy expended upon the contrivance (c/. Art. 36).

77. Efficiency. As the cost of the energy used in a manu-
facturing plant or a system of transportation is a very large part of
the operating expense, the efficiency of the machinery used is a
feature of great importance. It often proves to be very poor
economy to buy machinery of low efficiency simply because it is
cheaper.

Since some useless work is always done, no machine has an
efficiency of 100%. No machine can create energy (c/. Art. 36);
it can only transfer or transform energy that is supplied to it
from some external source. It enables the user to apply his energy
in more convenient ways than would be possible without it; but
the user is always taxed, as it were, a certain per cent of the energy
for the convenience thus obtained.

78. Mechanical Advantage from Law of Machines. The
law of machines enables us to find the mechanical advantage of
any machine. For we may write equation (7) (Art. 76) in the form

T — ^y i-^-> *^^ mechanical advantage of any machine is

equal to the ratio of the displacement of the effort to that of
the resistance. It is often more convenient to find the mechanical

PHYSICS

advantage of a composite machine by measuring these distances
than it is to calculate it by multiplying together the mechanical
advantages of the parts. This is the case with the screw.

79. The Screw. The thread of the screw is an inclined
plane wrapped around a cylinder. Fig. 49 shows how the screw
would look if one turn of the thread were
unwrapped.

In order to turn the screw about its
axis, a force /^ is applied at the end of the
lever, or at the circumference of the head;
and its displacement Z^, for one turn, is
the circumference described by the point
of application of this force.

When the screw is. rotated, either the
screw itself or the nut in which it turns,
moves in a direction parallel to the axis.
Thus when the jack screw (Fig. 50) is turned once around, the
stone, or whatever rests on the head of the
screw, is lifted through the distance /j be-
tween two adjacent turns of the thread.
This distance, measured parallel to the
axis, is called the pitch of the screw. Finally
if /a represent the resistance to be overcome.

Fig. 49. Screw Thread
Unwrapped.

have from equation (7), t ^ ^f which
/I h

tells us that the mechanical advantage
OF THE SCREW is nunierically equal to the
circumference through v* which the effort is
applied, divided by the pitch of the screw.
Since the lever or head of the screw may
b2 made very large, and the pitch very small,
this equation shows that the mechanical ad-
vantage may be enormous, and is limited
only by the strength of the materials used
in the construction of the screws. Thus, wagons, locomotives,
and even large buildings are lifted by means of jack screws.

Fig. 50. Jack Screw

MOMENTS

Fig. 51 shows how a large house was lifted up a hill 100 ft. high
with the help of such screws. They may be seen in the picture
between the timbers and the house.

80. The Equal Arm Balance, which is generally used for
comparing masses, is another important application of the law of
moments. If we wish to weigh a certain quantity of some sub-
stance, for example a pound of sugar, the mass, of the sugar in one

FiQ. 61. Jack Screws in Action

pan is assumed to be one pound when its weight balances a standard
pound weight on the other pan. For if the balance comes to
rest with the pointer at zero the opposing moments are equal.

Hence if /j represent the weight of the sugar, and /^ that of the
standard pound mass, and if r^ and r^ represent the corresponding
arms, then the equation for the balanced moments is ^rj = f^r^.
But r^ = r^, therefore /j = /i, i.e., the weights are equal. This will
be true if the arms are exactly equ^l, and if the balance comes to
rest with the pointer at zero under no load.

And since it was shown in Art. 31 that the weights are piopor-
tional to the corresponding masses, it follows that if the weights are
equal the masses are also equal. Thus the law of moments shows
us that we are correct in our habitual assumption that we can com-
pare masses correctly by the process of weighing. Of course the

90 PHYSICS

accuracy of the comparison is dependent on the accuracy of the
balance and of the masses in the set employed as standards.

8L Looking Backward. We have now arrived at a place
in our studies in Physics where it will be well to pause and
review what we have learned. The principles are really very
few^ although, as we have begun to see, their applications to every-
day life and to the devices of modem civilization are countless.

First, we have learned how uniform and uniformly accelerated
motions may be accurately and concisely described, and especially
so by the graphical and analytical Inethods. We then endeavored
to gain clear notions of the relations of mass and acceleration to
work, energy and activity, or power; and we found that by means
of concise equations in which these relations are expressed, many
important practical problems may be solved.

We then considered the behavior of bodies in motion and
of bodies in equilibrium when acted on simultaneously by two
or more forces; and we found that the resultant motions and
the resultant forces can be represented with great ease and clear-
ness by means of vectors. Further, we learned that many compli-
cated machines are made up by combining the principles of two or
more of the elementary mechanical devices known as the inclined
plane, the lever, the pulley and the screw; and that for each of
these devices a simple numerical relation between the effort and
the resistance can be established. This is done by compound-
ing or resolving forces or motions with the aid of vectors, or
by taking the moments of all the fojces with respect to some
conveniently chosen axis, and forming the equation for their equi-
librium.

In conclusion, we found that whenever energy is expended
upon any kind of machine for the purpose of doing work, the sum
of the useful work and the inevitable useless work is the exact
equivalent of the energy expended — no more, no less.

82. Our Future Study of the several forms of energy will
show us that there is nothing in the study of Physics but the accurate
description of relations that occur when energy is transferred
from one portion of matter to another,, or changed from one form

MOMENTS 91

into another fonn. We can gain knowledge of these changes only
through observation and experiment

Phenomena are thus learned and grouped into classes. The
conditions under which they occur and their relations to each
other are described in concise statements called laws. With
the aid gf the reasoning powers and the trained imagination,
HYPOTHESES are framed for the explanation of these laws. The
hypotheses are then tested by deducing from them relations which
follow as necessary consequences. Careful experiments are then
devised and carried out in order to determine whether or not
the relations thus deduced are verified — that is, whether they
are true or not.

When a hypothesis is found competent to explain every known
fact that must follow as a consequence of it, and is verified by
every appropriate experiment that is made in order to test it, it
takes rank as an established theory.

By deducing from a hypothesis or theory certain consequences,
and then testing these deductions experimentally, most of the great
scientific discoveries have been made.

The method of study here outlined is called the scientific
METHOD. Since the history of great scientific discoveries, and
of the inventions which have always followed in their wake, has
plainly shown that this method is the only one by which such
advances have been made, the great advantage of a study like
Physics is manifest. It is only by training the powers of observa-
tion and reasoning, and by developing the scientific imaginiation
in as many people as possible, that individuals can be produced
who shall continue the progress in discovery and invention which
is now going on. For discoverers and inventors must not only
be bom and educated, but they must be supported materially, and
encouraged by an intelligent interest on the part of the great body
of people among whom they live and work.

SUMMARY

1. In order that a body may be made to turn about an axis,
it must be acted on by a force whose line of direction does not
pass through the axis.

92 PHYSICS

2. The effectiveness of a force in producing rotation is its
moment. Moment of force = force X ann of force.

3. The mechanical advantage of a lever may be found by
equating the opposing moments taken with respect to the fulcrum.

T* • 1 * ru X- effort arm

It is equal to the ratio, — r-r .

^ resistance arm

4. The mechanical advantage of a lever is also equal to the

^ displacement of effort
* displacement of resistance'

5. The resultant of two parallel forces having the same direc-
tion is equal to their sum; it has the same direction as the two
forces; its point of application lies on the line joining theirs, and
divides that line into segments that are inversely as the magnitudes
of the two forces.

6. In order that any system of parallel forces may be in equilib-
rium, the sum of the forces in one direction must be equal to the
sum of the forces in the opposite direction; also the sum of the
moments tending to turn the system in one direction about any
point or axis must be equal to the sum of those tending to turn
it in the opposite direction about the same point or axis.

7. The equilibrant of any number of parallel forces may be
fully determined by means of equations formed in accordance
with this statement.

8. The center of mass of a body is the point of application of
the resultant of any set of forces that act in the same direction
equally on all its particles.

9. The center of gravity of a body is the point of application
of its weight, and is identical with its center of mass.

10. The stability of a body is measured by the work that must
be done in overturning it.

11. The mechanical advantage of a composite machine is equal
to the product of the mechanical advantages of all its elementary
parts.

12. In the case of every mechanical contrivance, the work
done by it is equal to the work done upon it, or f^l^^ = /jZj.

MOMENTS 93

13. The work done by every machine includes some useless
work.

14. The mechanical advantage of any machine is also equal

^, ^. displacement of the effort ,.,,.,

to the ratio, t^ r i-"j.^i ^i > both displacements

displacement of the resistance ^

being measured in the directions of their corresponding forces.

15. The mechanical advantage of the screw is equal to the
. circumference described by the effort

' pitch of the screw

16. The equal arm balance is used for comparing masses by
means of their weights.

QUESTIONS

1. How must force be applied to a body in order to make it rotate
about a given axis?

2. What is meant by the moment of a force with respect to a given
axis? What is its numerical measure?

3. Show how, by equating the moments about the fulcrum of a
lever, we can find the equation for its mechanical advantage.

4. Show by geometry that for a lever the displacements are pro-
portional to the corresponding arms. What, then, is the relation
between the displacements and the corresponding forces?

5. Show in the case of the lever that the work done by the effort
equals that done on the resistance.

6. What are the conditions that must be satisfied in order that any
system of parallel forces may be in equilibrium? What kind of motion
will result if each of these conditions is not satisfied? If neither is
satisfied?

7. Explain what the center of mass of a body is.

8. Explain why the weight of a body may be supposed to be a
single force, acting at its center of mass.

9. What is the measure of the stability of a body?

10. Show how the stability of a body may be determined graph-
ically.

11. Show by a diagram that the stability of a suspended body is
increased by increasing the distance between its center of mass and
its point of suspension; and vice versa.

12. Show by diagrams that the stability of a body supported on a
horizontal plane is increased (a) by increasing the area of the base,
(6) by lowering the center of mass.

13. Explain how the principles of stability are applied practically
m the construction and loading of buildings, wagons, and cars.

94 PHYSICS

14. Explain why a man on a step ladder is more easily overturned
the higher he ascends, unless the feet of the ladder are put propor-
tionately farther apart.

15. In what two ways may the mechanical advantage of a com-
posite machine be determined?

16. State the law of machines, and write the equation that ex-
presses it analytically.

17. What kinds of useless work are done by a machine?

18. Of what commercial importance is the efficiency of machines?

19. What are some of the uses of the screw? How may its mechan-
ical advantage be determined?

20. From the law of moments, show why we can correctly compare
masses by means of the equal arm balance.

PROBLEMS

1. A workman applies 75 Kg-force at one end of a crowbar 200 cm
long; what weight may be lifted at the other end distant 25 cm from
the fulcrum? What is the mechanical advantage? With the same
ratio of the arms, what effort will be required to overcome a resistance
of 800 lb.? In each case what was the amount and direction of the
pressure exerted by the fulcrum?

2. A safety valve lever, Fig. 52, has its fulcrum at a, and is to push
down on the valve rod at c, which is 2 cm from a. What must be the

weight of the ball, if it is to be applied at
notch 5, which is 6 cm from a, and exert
at c a 5 kilograms-force? What force will
be exerted at c if the ball weighs 2 Kg and
is placed at notch 10, which is 12 cm from
o?

3. Suppose that a trunk 0.9 m long, 0.6
m high, and weighing 120 Kg is to be tipped
over on one end by lifting at the other. The weight is uniformly dis-
tributed: how much force is necessary to start it? Will this force
increase or diminish as the trunk approaches the upright position? Rep-
resent graphically the stability of the trunk, and express the value
of the stability in kilogram-meters of work.

4. Devise a scheme for weighing a turkey, using only a stick of
uniform density and cross-section, and 50 cm long, a cm scale, some
strong cord, and a flatiron known to weigh 2.73 Kg (6 lb.). Be sure
that your scheme provides for eliminating the weight of the stick.
Illustrate your method by working out a numerical example.

5. A bridge whose weight is 3 X 10^ lb. rests on abutments 75 ft.
apart. Assuming that the weight of the bridge is uniformly distrib-
uted, what part of this weight is supported by each abutment? What

MOMENTS 95

additional pressure is applied when an engine that weighs 12 X 10*
lb. stands with its center of mass 25 ft. from one end of the bridge?

6. With a single fixed pulley (Fig. 53) , what pull on one end of the
cord will support a weight of 100 Kg at the other? Use the lever
principle, assuming the radius of the pulley to be 5 cm.

In this case what is the pull on the support if the pulley i "^ i

itself weighs 2 Kg? When the resistance is overcome * — h ■ '

through 1 m, through what distance does the force act? V

What is the mechanical advantage? x-

7. A movable pulley is arranged as shown in Fig. 54. — '

With what force /i, and in what direction, ^
J I must you pull in order to support a weight ,

^y^ /j, of 150 Kg? Would the force be the same t
I if you pulled at /, making use of the fixed
"^ pulley? Use the lever principle, taking q p

for one fulcrum and i for the other. Can you V y
obtain th^ same result from the law of ma- j ^

chines [equation (7), Art. 76]? Express the
mechanical advantage in terms of the lever
Fig. 53 arms, and also in terms of the distances. ^iq. 54

8. Show that if the arrangement of pulleys
in Fig. 54 were turned end for end, the mechanical advantage would
be 3 for a pull at /. In arrangements of this sort, what is the relation
between the mechanical advantage and the number. of parts of the cord
that pull against the resistance? Is there any useless work done by
the pulleys, so that the mechanical advantage actually obtained is less
than that given by the calculation?

9. The screw of a cider press has a pitch of 0.5 cm, and is turned
by a lever 50 cm long. What is its mechanical advantage? What
pressure will be exerted on the apples when you apply a 30 Kg-force at
the end of the lever?

10. Make a diagram of a combination of any two of the machines
mentioned in this list of examples. Find the mechanical advantage
of the combination (c/. Art. 75), and the effort necessary to overcome a
resistance of 400 Kg-force.

SUGGESTIONS TO STUDENTS

1. Measure the lever arm and the pitch of the screw of a vise, and
find its mechanical advantage.

2. If you are interested in turning lathes, examine one in a shop,
and see how many of its parts have mechanical advantages. How is
more force and slower speed obtained by shifting the belt from one
pair of pulleys to another? When a screw is to be cut, how do you

96 PHYSICS

make the cutting tool travel along the lathe bed at the desired rate; —
for example, to cut twice as many threads to the inch as there are
on the lead screw? What other examples of the composition of mo-
tions and of the lever principle does the lathe furnish?

3. What kinds of lever can you find in a sewing machine? in a
typewriter? in a bicycle? Consult a book on physiology and see if
you can find the lever principle in the human arm, foot, jaw, etc.

4. Can you solve the lever problems presented in rowing a boat?
in using a nut-cracker? in the sugar tongs? in the scissors? in the gas
tongs? in the claw hammer?

5. With a set of pulleys like Fig. 54, determine by experiment the
number of gms- weight at / that will just lift a given weight at W with
uniform speed. Measure the distance I through which / moves, while W
is being lifted a distance /i = 10 cm. Calculate / X ^, the work done by
the effort, and W y^h, the work done on the resistance; also calculate

the efficiency, . , . Now, by taking off weight at /, find the number

of gms-wt at / that will just allow W to descend with uniform speed;
and also find the efficiency in this case as you did in the first. Take the
average of these two efficiencies as the mean efficiency for the given load.
In the same way, find the mean efficiencies for, say, 9 other loads, and
choosing a convenient scale, plot a graph with efficiencies for ordinates,
and loads for abscissas. Does the efficiency increase with the load?
In direct proportion, or according to some other law? A set of pulleys
can be bought cheaply at a hardware store, and will be all the more
interesting if not too good. How can you determine the number of
gms-force of the friction?

6. If mechanically inclined, you may find mines of interesting infor-
mation about all sorts of mechanical devices, their mechanical advan-
tages and efficiencies in Perry's Applied Mechanics (Van Nostrand,
N. Y.), and in Pullen's Mechanics (Longmans, N. Y.). There is much
in these books that you may not be able to understand; but you can
read without difficulty enough to increase immensely the knowledge
that you have thus far acquired. Lodge's Mechanics (Macmillan,
N. Y.) is easier reading, and will also interest and help you.

-3
M

w
w

CHAPTER V

ROTATION

Note. The authors recommend that this chapter be used for informal
dbcussion on the first reading. If time is limited it may be omitted.

83. Flywheels. In the preceding chapter we learned that, in
order to cause rotary motion, an unbalanced moment of force is re-
quired.

Another important case of the conversion of translatory motion
into rotary motion is that of a stationary engine and its flywheel,
Plate III. Here the relations involved in producing the rotary
motion are the same as in the locomotive and its drivers. But the
flywheel is large and has a very massive rim, and is designed to
produce an effect which is not necessary in the locomotive drivers.

This effect is that of steadying the motion; for it is clear from
what has preceded that the moment of force acting on the wheel
is different in different positions of the crank pin, and this will
cause a jerky motion of the machinery. But the big flyvv^heel
receives and stores up energy of rotation when the crank pin is in
the favorable positions, and faithfully pays it back again when the
crank pin is in the unfavorable positions. Thus it prevents the
sudden jerks which would be injurious to both the engine and the
machinery which it runs.

Why is it that the flywheel has a massive rim and large
diameter? How does this distribution of the mass make it more
effective in storing up energy and paying it out again?

84. Angular Measures. These questions can not be answered
by expressing the relations in terms of linear velocity,
because it is clear that different portions of the mass, being at
different distances from the axis of rotation, have different linear
velocities Therefore we must have some other means of measuring
this velocity. Now, it is evident that every spoke of the wheel, and

98 PHYSICS

in fact every radius, sweeps over the same angle in the
same time, and therefore all the particles of the wheel have
the same angular velocity. How, then, is angular velocity
measured?

A convenient way to measure an angle is to find the number
of times that the radius is contained in the corresponding arc; i.e.,

. _ length of arc
^ length of radius'

If in this equation we make length of arc equal to length of

radius, we have, angle = — = 1. Hence, the appropriate unit

angle is that angle which corresponds to an arc whose length equals
that of the radius. This unit angle is called the
RADIAN, and by a simple calculation is found
yN. to be equal to 57°.27, Fig. 55.

/ \ It should be noted that since the numer-

/ \ ical value of an angle is simply the number of

f \ times that the radius is contained in the cor-

'^^ 1 responding arc, it is not expressed in grams, or

Fio. 56. One Radian centimeters, or seconds; and hence angle has
no symbol in terms of these fundamental units.
Now, since the angular velocity is the ratio of the angular space
described to the time in which it is described, and since the unit
angle is the radian, angular velocity is measured in radians per
second; and unit angular velocity is the angular velocity of a body
which rotates at the rate of one radian per second. Since unit
angle = 1, the symbol for unit angular velocity is ^.

If the angular velocity varies, there will be a rate of change of
angular velocity, i.e., an angular acceleration; and the measure
of this angular acceleration is, of course, the change in angular
velocity per second. Since

, , .. angular velocity

angular acceleration = — ^ — sL

^ time

the unit angular acceleration is one per second, and its

° sec ^

symbol is ^,.

ROTATION

85. What Corresponds to Mass? We have now learned
that when we are deaUng with rotation instead of translation, we
must consider moment of force instead of force, angle instead of
distance, angular velocity instead of linear velocity, and angular
acceleration instead of linear acceleration. But what, in the
former case, corresponds to mass in the latter?

The answer to this question may be obtained by considering
the case of a small boy swinging on a gate (Fig. 56). Suppose
that the gate is open, and that a boy is perched on it at a distance
of 100 cm from the hinge, or axis of rotation. For simplicity let
us leave out of consideration the moment of force necessary >to close

Fig. 56. A Moment of Force Produces Angular Acceleration

the unloaded gate, and ask how much must be the extra moment
of force necessary to give the gate a certain angular acceleration
when there is a second child on the gate with the first (Fig. 57).
If the masses of the two children are equal, then, since all the condi-
tions are the same as before except that the mass has been doubled,
it is evident that the required moment of force must be twice as
great for two children as for one, three times as great for three
children as for one; and so on.

Hence, in general, it appears that when the arm of the mass is
constant, the moment of force necessary to impart to a mass a given
angular acceleration is directly proportional to the mass.

Again, let us ask how the moment of force required to give the

100

PHYSICS

gate a certain angular acceleration is affected by a change in the
distance bf the boy from the axis.

Suppose that the one boy is now 200 cm instead of 100 cm
from the axis (Fig. 58). The required moment of force will now be

FiQ. 57. Greater Mass : Greater Moment op Force

much greater. But how much greater will it be? The mass of
the boy now has twice the arm, and therefore his moment, with

Fig. 58. Moment op Force is Proportional to (Arm op Mass)*

respect to the axis, is doubled, and the moment of force required
to produce the given angular acceleration would be doubled on
this account alone. But since the mass must now move through
an arc twice as long in the same time as before, it follows that

ROTATION ^ 101

the moment of force required wouM-'b^'doubled for this reason
as well. Therefore this moment must Be.2; J< ^ = 2* = 4 times as
great as before. Similar reasoning shows*'>tl4*^ i^ the boy were
300 cm from the axis the required moment jis'S^X 3 = 3^ = 9
times as great as when the boy is 100 cm from ihe,s^&i6p^ In gen-
eral, then, we find that the moment of force required' ;ta'jtroduce
the given angular acceleration is proportional not only to the n^iss,
but also to the square of its distance from the axis. -I^ ..\.,

Furthermore, if we wish to shut the gate more quickly, "a^j
other conditions remaining the same, a greater moment of force •
is required to do it. It can be shown by experiment that the
moment of force must be increased in the same proportion as is the
angular acceleration that it is to produce.

Therefore, since the moment of force required to put the mass
into rotation about an axis is directly proportional to the mass, to
the square of the distance of the mass from the axis, and to the
angular acceleration; and since it depends on these quantities
alone, it follows that we may write as the equation for rotary motion,
moment of force = mass X (arm of mass)^ X angular acceleration.

Since for translation

force = mass X linear acceleration,
we see that the quantity that stands in the same relation to moment
of force and angular acceleration as does mass to force and linear
acceleration is mass X (arm of mass)^.

This quantity is called the moment of inertia of the mass
about the given axis and is generally denoted by I.

86. Rotation vs. Translation. These relations which we have
.only roughly illustrated in the case of the boys on the gate have
all been verified by careful experiments and shown to be true in
all cases; so that in general when we are considering

ROTATION instead of translation, we must consider

Moment of force " " Force,

Angle " " Distance,

Angular velocity " " Linear velocity,

Angular acceleration " " Linear acceleration,

Moment of inertia '* " Mass.

102 ^ PHYSICS

Furthermore, we may.obtain the relations that hold in cases of
rotary motion from^tmlse for the corresponding cases of translatory
motion simply .bj'.^king the substitutions according to the table
just given. ^.Eof*^xiample, we have learned (c/. Art. 39) that the
kinetic eiieygy«6f a body in translatory motion is equal to

%/;•* mass X (velocity)*

2 ■

c-^Jcence the kinetic energy of a body in rotary motion is equal to

moment of inertia X (angular velocity)*

87. Determination of Moment of Inertia. The determi-
nation of moment of inertia is usually a difficult problem, because
different particles of the mass are generally at different distances
from the axis, so that the quantity (arm of mass) is different for
different particles. Hence, in order to get the value of the
moment of inertia of any rotating mass, we must first consider
the moments of inertia of the single particles and then sum
them all up.

Thus for the rim of the flywheel. Fig. 59, the moment of inertia
of a particle on the outside of the rim is the mass of this particle

multiplied by the square of the outer
radius of the wheel, and therefore that
of the layer of particles in the outer rim
is the mass of that layer multiplied by
the outer radius squared, because the
arm of each of these particles is equal
to this radius. Likewise the moment
of inertia for the layer of particles in
the inside of the rim is the total mass
of the particles in that layer multiplied

Fig. 59. Moment op Inertia , ., - ., ,. <. .i •

Equals Mass X (Radius op by the square ot the radms oi the in-
side of the rim. Now the remainder
of the mass of the rim may be conceived to be made up of similar
layers of particles, with radii that are intermediate in length be-
tween the inner and the outer radius.

ROTATION 103

Accordingly the total moment of inertia of the rim is the sum
of all the products obtained by multiplying the mass of each layer
by the square of its radius. But since the sum of the masses of
the layers is the total mass of the rim, and since the radii are all
intermediate in value between the outer and the inner radius,
there must be some radius intermediate in value between the
inner and the outer such that multiplying the total mass
by the square of this intermediate radius will give the same result
as would the summing up of all the separate products obtained
by multiplying the masses of the several layers by the squares of
their respective radii. This intermediate radius is called the

RADIUS OF GYRATION.

The determination of the total mass of the rim is an easy
geometrical problem, but the calculation for the radius of gyration
requires the higher mathematics. In cases like that of the rim
of the flywheel, unless great accuracy is required, we may assume
that the radius of gyration is equal to half the sum of the inner
and the outer radii of the rim.

88. Effectiveness of Flywheels. The effectiveness of the
flywheel in steadying the motion depends on the magnitude
of its moment of inertia, and since this magnitude is proportional
not only to the mass of the wheel, but also to the square of the
radius of gyration, it becomes apparent that not only must the
mass be large, but that also this mass must be placed as far as
practicable from the axis. In fact, we see that a wheel of large
radius of gyration is just as effective as one of half the radius of
gyration and four times the mass.

From this it might seem that we could increase the effectiveness
of the flywheel indefinitely by increasing its radius of gyration
without correspondingly increasing its mass. There is, however,
a limit which can not be passed with safety, because the flywheel
may burst. A little careful reasoning with the assistance of our
algebra and geometry will enable us to see why this is true, and
also to arrive at a very important general principle.

89. Conditions for Circular Motion. Let the circle p c q,
whose center is o. Fig. 60, represent a circle on the rim of the fly-

104

PHYSICS

wheel, and p the position of a small portion m of its mass at a
given instant. Suppose that the wheel is rotating so that the
linear speed along the arc is uniform.

According to the first law of motion, m, if not a part of the
rigid wheel, would move in the direction of the tangent to

the circle at p; but since it is con-
strained to remain on the arc, it
will at the end of a very short time t
arrive at some point c on the circum-
ference. Since t is very small, the
arc pc will be so nearly equal to the
chord that we may without appre-
ciable error regard the arc and chord
as identical. We may now let pc
be the vector that represents the mo-
tion along the arc. As in the case
of the inclined plane, we may re-
solve this motion into two components, one in the direction of the
tangent and the other in that of the radius po. The lines pb and
he will be the two component vectors.

Now if we extend the radius po to cut this circumference at
q, and draw qc, the right triangle, pcq and pbc are similar (Why?);

Fig. 60.

The Acceleration to-
wards THE Center is —
r

therefore we have —

be pc \ , pe^

— = ^-— ; whence, be = ^—,
pe pq pq

But be repre-

sents the distance that m, starting at p, traverses in the direction

af , , .

po in the small time t; and hence it is equal to — , in which a is

the linear acceleration of m in that direction. Also pe represents

the distance that m traverses in the time t with the uniform velocity

along the arc pe\ so that if v represent this velocity, pe = vt (Why?).

Further, pq is 2r, i.e., twice the radius of the circle. Therefore,

when we substitute these values in the foregoing equation, we have

af v^f

— - = -^. Solving this for a, the acceleration toward the center.

we obtain a = —

The centrally directed force (sometimes called centripetal

ROTATION

105

force) that causes this acceleration — must be [cf. equation (4),

It means that a mass,

Art. 27'\ f = ma = — . This force, of course, must have an

equal and opposite reaction, which is often called centrifugal
force.

The conclusions expressed by this equation will follow for
any other small time t and for any other point of the circumfer-
ence: hence the equation

' ~" r
applies to all cases of rotary motion.
m order to move with
uniform linear speed
around the circumference
of a circle, must be acted
on by a constant force
whose direction is always
toward the center, and
whose magnitude is di-
rectly proportional to the
mass and the square of the
linear speed, and inversely
proportional to the radius.
If m is expressed in

S°^''^^^I^' ^^^ ^^" ^"^'
then / will evidently be in

FiQ. 61.
The Centripetal Force is

Loop the Loop

gm y cm2
cm ^ sec2

^^^, which will be recognized as the symbol for

dyne.

'90. Why Wheels Burst. It is often convenient to express
this relation in terms of angular units instead of linear units. In
order to do this we must substitute angular velocity for linear
velocity and moment of inertia for mass, according to the table,
Art. 86. Thus, if u represent the angular velocity

. _ mv^
' r

mr^u^

= mrur.

(8)

106

PHYSICS

This equation shows that if we increase the radius of a fly-
wheel indefinitely, while the angular velocity u and mass m remain
the same, the force required to hold the parts together will soon

become greater than the cohesive force
of the particles and the wheel will then
burst.

The equation also shows that for a
given wheel, for example, an emery wheel,
Fig. 62, the force required to hold the
parts together increases as the square
of the angular velocity u\ and hence if
we continue to increase the number
of revolutions per second of the wheel,
a limit will be soon reached beyond
which the angular velocity can not be in-
creased with safety.

Fig. 62. The Emery Wheel
MAY Burst

91. Distribution of Mass. There
is another very important condition
which must be observed in the construction of a flywheel, or in
fact of any other rotating part of a machine. The equation
/ = mrt^ tells us why this condition must be complied with,
for it is evident that for a given body rotating about an axis,
all the small masses m have the same angular velocity u) and hence
the centrally directed force required to keep each such mass moving
in its circle, depends on the value of the quantity mr corre-
sponding to this mass. This latter quantity mr or mass X (arm of
mass) is often called moment of mass, just as force X (arm of
force) is called moment of force. Now, unless the central force for
any mass m on one side of the axis be balanced by an equal and
opposite force on the opposite side of the axis, there will be art un-
balanced lateral strain on the axis, and the wheel will wabble.
Hence it is clear that the condition that must be fulfilled to
prevent the wabbling is that every moment of mass on one side^
of the axis must be balanced by an equal moment of mass on the
opposite side. In other words, the moments of mass must be symmet"
rically disposed about the axis (cf. Art. 74).

ROTATION 107

92. Bailroad Cnrves. Since the relation / = — , which

we found to hold in the case of the flywheel, applies to every mass
that is moving with uniform linear speed in a circular path, it must
apply to the case of a railway train when it rounds a curve.
Railway curves are usually arcs of circles, and the portions of
straight track at the two ends of the arc are tangents to the circle
of which the curve is a part.

In accordance with the first law of motion, the car, at the instant
when it reaches the curve, tends to continue moving with uniform
velocity in the direction of the tangent. For have not all of
us had the experience of being apparently thrown against the
side of the car when it began to round a curve unexpectedly?
The car turns out of its straight path because it is pushed laterally
by the track; but the passenger continues in the straight path
until he is suddenly pushed into the new, direction by the side of
the car. It is clear, then, that to keep the car moving in the curve,
a horizontal force must continually be exerted by the track; and
that this horizontal force must act so as always to be perpendicular
to the track at the point where the pressure is exerted. This
lateral force exerted by the rails is evidently directed inward toward
the center of curvature, i.e., along the radius at the point where
the push is exerted; and, as we have seen in the case of the rim of

the flywheel, it must be equal in magnitude to , where m is

the mass of the car, v the uniform speed along the curve, and
r the radius of curvature of the track. Now, how must the track

be built that it may exert the central force with the minimum

strain on both itself and the train? This is accomplished by
inclining the road-bed so that the outer rail is higher than the
inner.

. Let xyz^ Fig. 63, represent a cross-section of the roadbed
and c the center of mass of the car, where all the forces may be
supposed to act. Since the pull of the engine is balanced against
the friction and air resistance^ these forces may be left out of the
problem. The forces that must be supplied by the track are,
first, a force vertically upward and equal to mg, the weight of the

108

PHYSICS

train; second, a force horizontally inward toward the center of
mi?

curvature and equal to ■

These forces are represented by the

vectors hp and pc respectively; therefore the resultant F of these

two forces is represented by the. vector
he. Now, in order to exert this force
with the least possible strain on the
rails, the roadbed must be perpendicu-
lar to he. When this is so, we see from
the geometry of the diagram that

FiQ. 63. The Curved Track
IS Inclined

zy pc r V" , ^,
— ^ = f- = ^ = — , I.e., the necessary
xz op mg rg ''

lateral slope of the track is directly pro-
portional to the square of the velocity
and inversely proportional to the radius
of curvature. Furthermore, it is not
affected by the mass of the car, since the
mass cancels out of the expression.
The student will recall the similar cases of the inclination of
a circus ring or race track and the inward leaning of the horse
and rider; also the impossibility of mak-
ing a turn with a bicycle on a slippery
pavement without slackening speed.

93. Spinning Tops. Another inter-
esting and important fact about rotary
motion is that the inertia of a rotating
body shows itself not only in the tend-
ency to continue rotating when started,
but also in the resistance which it offers
to any force tending to change the direc-
tion of its axis of rotation. This is well
illustrated in the case of the top. Fig.
64, which will stand on its point, and resist any force tending to
overturn it, only so long as it is rapidly spinning. A similar case
is that of a rifle ball, which is given a rapid spin about its long-

FiG. 64. The Top Stands
ON ITS Point only while
Spinning

ROTATION 109

est axis by cutting helical grooves in the rifle barrel. Since this
longest axis of the projectile coincides with the path in which it
was started, the bullet tends to continue in its path point fore-
most, and to strike in that attitude.

SUMMARY

1. The unit angle is the radian; it hs^s no symbol in terms
of gm, cm, and sec.

2. The value of the radian in degrees is 57° .27.

3. The unit angular velocity is one radian per second. Its
symbol is ^.

4. The unit angular acceleration is one radian per second
per second. Its symbol is ^.

5. The moment of inertia of a rotating particle is mass X
(arm of mass)*.

6. The relations of moment of inertia to rotation are the same
as those of mass to translation.

7. The equations of rotary motion may be obtained from
those of translatory motion by substituting moment of force for
force, angle for distance, angular velocity for linear velocity, angular
acceleration for linear acceleration, and moment of inertia for mass.

8. The moment of inertia of an extended mass is equal to the
sum of the moments of inertia of its separate particles.

9. The numerical value of the moment of inertia of an extended
mass may be obtained by multiplying the total mass by the square
of the radius of gyration.

10. A body will not move in a curved path unless it is con-
stantly acted on by a central force that gives it a uniform accelera-
tion toward the center of curvature of the curved path.

11. The numerical value of the necessary acceleration toward

the center in linear units is — , and in angular units it is rv*.

12. The force / that will keep a body moving uniformly in a
circular path is equal to .

13. Moment of mass is mass X (arm of mass).

14. In order that a body may rotate smoothly about an axis.

110 PHYSICS

the moments of mass of all its particles must be symmetrically
disposed with respect to that axis.

15. A rotating body resists any force tending to change the
direction of its axis of rotation.

QUESTIONS

1. What is the use of the flywheel of a stationary engine?

2. What is meant by the angular velocity of a rotating body? Is
it the same for all particles of the body?

3. What is meant by the radius of gyration of a rotating body?
How can the moment of inertia of an extended mass be calculated
when its radius of gyration is known?

4. Why is it important in the case of a flywheel to have a large
radius of gyration? How is the wheel made so as to secure this result?

5. How may material be economized in the construction of such
a wheel, and why?

6. Explain why a flywheel or an emery wheel will burst if of too
large diameter, or if rotated too rapidly.

7. With the aid of a vector diagram, show why a curved railway
track must be inclined inward toward the center of curvature. Give
some examples of similar cases.

8. Mention two ways in which the inertia of a rotating body mani-
fests itself, and illustrate by examples.

PROBLEMS

1. Since a circumference = 2ir X radius, 360° = how many radians?
How many degrees are there in one radian?

2. An emery wheel makes 2400 revolutions per minute; what is its

I 1 'J. ' radians^
angular velocity m ?

3. A moment of force whose average numerical value is 4 X lO**

gives to the flywheel of an automobile an angular acceleration of
rfl-fiijins •*

2 5—. What is the moment of inertia of the wheel? How many

sec^ ,. -^

•po li 1 f) Tl Q

seconds are required to give this wheel an angular velocity of 20 ?

4. The rim of a flywheel has a thickness of 20 cm, an inner radius
of 90 cm, and an outer radius of 110 cm. What is its volume? If
its density is 8, what is its mass? Taking the mean radius as the radius
of gyration, calculate the moment of inertia of the rim.

5. How many dynes must act with an arm of 20 cm in order to

give the wheel of problem 4 an angular acceleration of 1 ?

6. An emery wheel of 15 cm radius is making 20 revolutions per

ROTATION 111

second. How many dynes are required to keep each gram of emery
at the circumference from flying off?

7. In a loop-the-loop, Fig. 61, when the car is at the top of the
loop, what will happen unless the centripetal force necessary to keep
the car moving in the circular path is equal to or greater than the
weight of the car? Let m represent the mass of the car, 980 the accel-
eration of gravity, v its linear velocity at the top of the loop, and r the
radius of the loop, 'and show that the car will not fall if m X 980 =

— . Need the mass be considered? If the radius of the loop is
r

500 cm, what must be the velocity r?

SUGGESTIONS TO STUDENTS

1. See if you can swing a small pail full of water around in a ver-
tical circle without spilling the water. Is this experiment like a loop-
the-loop? If your arm is 75 cm long, what must be the number of
revolutions per second when the water does not spill?

2. Can you find out how the governor of a stationary steam engine
works? What can you find out from the laundryman about centrif-
ugal drying machines?

3. If you have occasion to take the rear wheel off your bicycle,
hold it by the step, spin it rapidly, rest the end of the step on your fin-
ger, and see what the wheel will do.

4. For interesting information about tops, gyroscopes, and rotation,
consult Hopkins, Exj)erimental Science, (Munn & Co., N. Y),
pages 10-37.

CHAPTER VI
FLUIDS

94, Pnmps. In the foregoing chapters, we have found it both
convenient and interesting to learn some of the principles of
Physics, and some of the methods of investigating physical
phenomena, by considering the motion of a railway train. We
found, that in order to answer only a few of the questions which
naturally arise in connection with the motions of a locomotive
and its parts, it was necessary to master much of that part
of Physics which deals with forces and their effects, and is called
Mechanics. When we take up the study of that form of energy
called Heat, we shall have frequent occasion to refer to the steam
engine; and in fact, if we wished thoroughly to understand the work-
ing of every part of the highest type of modem locomotive, and
explain all the physical phenomena that occur in connection with
it, we should find before we had finished that we needed to know
the greater part of what there is to be known about Sound, Light, and
Electricity, as well as about Mechanics and Heat. But, as we are
now to take up the study of the Mechanics of Fluids, we shall find
more obvious relations in that class of machines called pumps;
because the specific purpose of every pump is to propel some liquid,
like water, or some gas, like air or illuminating gas, and to deliver
it under pressure at places where it is to be utilized. Furthermore,
we shall gain a much wider view of the value-of such knowledge and
training as the study of Physics can give us, by finding out some-
thing of a few other great inventions that contribute largely to our
modem civilization, and are as closely related to our everyday lives
as is the locomotive engine.

In Plate IV is shown one of the powerful pumping engines that
distribute the water supply of a great city; and Fig. 65 shows a
very similar machine designed to distribute compressed air for
operating drilling machines and other appliances used in mines and

112

Plate IV. Pump in the 39tii St. Station, Chicago

This Pump has a capacity of 25,000,000 gallons a day. The pipes In front of the

picture contain the valves, the plungers for pumping are in the

cylinders farther in the rear.

FLUIDS 113

factories. These mammoth pumps grew out of very small be-
ginnings and were many centuries in coming to their present state
of power and efficiency.

95. Lift Pump. The common lift pump was known and used
in the time of Aristotle. The diagram, Fig. 66, shows how it is con-
structed and how it acts. It consists of a hollow cylinder at the bottom
of which is a valve opening upward like a trapdoor and called the

II IfHLlI

Figure 65. Air Compressor

inlet valve. Fitting closely into the cylinder is a piston perforated
by a hole over which is fitted another valve, also opening upward
and called the outlet valve. A long pipe called the suction pipe
extends into the water below. When the piston is lifted by means
of the piston rod, the outlet valve remains closed, while the inlet
valve opens, and air from the suction pipe enters the cylinder.
When the piston is pushed down, the inlet valve closes, so that when
the piston tends to compress the air in the cylinder, this air, by its
reaction, opens the outlet valve and passes through the perforation
in the piston. At the end of the stroke the air that had entered the

114

PHYSICS

cylinder is above the piston and
is lifted out by the piston during
the next stroke The next few
strokes remove the remainder of
the air from the suction pipe; and
the water which follows the air up
the pipe and into the cylinder,
passes through the pump and is
pushed out in precisely the same
manner as was the air.

But what force is it tliat
pushes the air and the water into
the suction pipe, and causes it to
lift the inlet valve and flow into
the cylinder? Aristotle and his
* followers offered in explanation
the saying that Nature abhors a
vacuum; but the reader will rec-
ognize that this is not an expla-
nation at all.

96. Force Pump. During the first century a.d., the force
pump (Fig. 67) was invented by a philosopher named Ctesibius of
Alexandria. This differs from the lift pump
only in that the piston is not perforated;
and the outlet valve is at the end of the
cylinder near the inlet valve. The outlet
valve must of course open outward. Ctesi-
bius also invented a double acting force
pump for putting out fires, which differed
but little from the hand fire engines now
used in villages. He knew a great deal
about how pumps worked; but discovered
nothing that enabled him to explain why
they worked. The idea that Nature ab-
horred a vacuum seemed sufficient to satisfy
the minds of most men until the seven-
teenth century, when there arose a most fiq. 67. Force Pump

Fig. 66. The Lift Pump

FLUIDS 115

remarkable group of men, who in their search for truth about
the things of the material world began to use the scientific
method (cf. Art. 82). The result was that they inade more
discoveries and contributed more to accurate scientific knowl-
edge in a few years than did all the philosophers in all the
centuries before them.

97. Air Has Weight. Galileo had proved that air has weight
by weighing a glass globe, forcing more air into it, and weighing it
again. The difference between the two weights, he rightly ascribed
to the weight of the air that had been added. He did not discover
that the weight of the air had anything to do with Nature's alleged
horror of a vacuum. He was astonished when informed that a
lift pump had been made with a suction pipe about forty feet
long, and that no amount of pumping would cause the water to rise
higher than about thirty-three feet. Since a vacuum remained in
the cylinder and upper part of the suction pipe, he was led to remark
that the horror of a vacuum was a force that had its limitations and
could be measured by the column of water that it would raise.

Galileo's friend and pupil Torricelli (1608-1647), who succeeded
him as professor at the Academy of Florence, took advantage of
this suggestion, and began a series of experiments which led him
to infer that the weight of the water column in the suction pipe was
supported by the weight of the atmosphere that rested upon the
surface of the water in the cistern." Reasoning that since mercury
is 13.6 times as dense as water, the weight of the atmosphere ought
to be sufficient to balance that of a column of mercury only
about one-fourteenth as long as the water column, he caused
two of his pupils to carry out the experiment, which is known by
his name.

98. Torricelli's Experiment. A glass tube, about 33 inches
long, was closed at one end and completely filled with mercury.
When the open end of the tube had been closed by the finger, and
the tube inverted, it was supported in a vertical position with the
open end in a dish of mercury. On removing the finger, the mer-
cury sank down a little way in the tube, and, after a few oscillations,

116 PHYSICS

came to equilibrium with the surface of the mercury inside the
tube about 30 inches (76 cm) above that of the mercury in the dish.
In the upper end of the tube was a very nearly perfect vacuum.
Torricelli noticed that the height of the mercury column often
varied; and he inferred that the variations were due to the changes
in the pressure of the atmosphere which was "now heavier and
dense, now lighter and thin."

Torricelli^s hypothesis as to the pressure of the air was thus con-
firmed so far as his experiments went, but other experiments
were necessary in order to establish its truth.

99. Pascal. When Pascal (1623-1662), who had been studying
the phenomena of fluids in equilibrium, learned of Torricelli's
experiments he repeated them, and concluded that "the vacuum is
not impossible in Nature, and she does not shun it with so great
horror as many imagine." Pascal reasoned that if one were to
ascend a mountain, the pressure of the air at the greater elevation
should be less, because there would be less air overlying the moun-
tain top than there was overlying an equal area of the plain. Ac-
cordingly he wrote to his brother-in-law, who lived near the Puy de
Dome, an ancient volcano in the Auvergne, France, asking him to
ascend the mountain with a Torricellian tube and observe whether
the mercury column would not fall because of the diminished
atmospheric pressure. The experiment was made; and it was found
that the mercury column became three inches shorter during the
ascent, but gradually resumed its previous length during the descent
to the plain.

Pascal also repeated Torricelli's experiment with wine instead
of mercury; and he found as he had inferred, that, since wine is less
dense than water, the atmosphere balanced a column of it which was
longer than the water column; for of course it would take a longer
column of the lighter fluid to make the same weight.

The hypothesis of Torricelli and Pascal as to the pressure of the
atmosphere was thus placed upon a firm experimental basis, and
was now competent to explain the phenomena of pumps; but it
required the evidence of many more experiments to secure its gen-
eral acceptance.

FLUIDS 117

100. The Mercurial Barometer. The mercurial barometer
which is an instrument of great precision, and of inestimable value,
is simply a Torricellian tube in which the dish for the mercury is re-
placed by a flexible bag of chamois skin. The tube and bag are
enclosed in a metal case which is fitted with a very accui^ate scale
by means of which the height of the mercury colunm may be
measured.

Since changes in the weather are caused by the passsing of areas
OF LOW PKESSURE, the barometric column falls when one of these
areas is approaching, and rises again after the low pressure area has
passed and a high pressure area has taken its place.

By means of barometers and other instruments, read simul-
taneously at scores of stations, the U. S. Weather Bureau officials
are able to map the weather conditions of the entire country
every eight hours; and thus, as the areas of low or of high pressure
travel across the country, taking with them their characteristic
weather conditions, the forecast official announces by telegraph the
probable time of its arrival and the kind of weather that may be
expected to accompany it. These weather forecasts and storm
WARNINGS, which would be impossible without the barometer and
thermometer, save many lives and much property annually. .

The barometer is also much used in measuring elevations,
such as the heights of mountains and the altitudes attained in
balloon ascensions. Near sea level, the barometer falls 0.1 inch,
or 2.54 mm, for every 80 feet of elevation; but at greater elevations,
since the density of the air is much less, the change of elevation corre-
sponding to a barometric depression of 0.1 inch is greater than 80
feet, and increases steadily with the increasing elevation. The
reason why the upper layers of atmosphere are less dense than the
lower layers is that those upper layers have much less air above
them pressing down upon them. With a good barometer a differ-
ence of four feet in altitude can be detected.

Since the pressure of the atmosphere on 1 cm^ exactly balances
the weight of a column of mercury having a certain length and
equal cross-sectional area, we can calculate this pressure in grams
or dynes per square centimeter by calculating the weight of this
column. Thus, at sea level, the average height of the bar-

118

PHYSICS

rometer column is 76 cm, and the density of mercury at 0° Centi-
grade — ^the freezing point of water — is 13.59 ^. The volume of
a mercury column 76 cm high and 1 cm* in sectional area is 76 cm'.
Its mass, therefore, is equal to the product of its volume and its
density, i.e., M = FD = 76 X 13.59 = 1032.84 gm.

Its weight in grams is therefore represented by the same number.
The average pressure of the atmosphere at sea level is thus found to
be 1032.84 ^^,.

Let the student substitute 1032.84 for m in equation (4), Art.
27, and 980 for a, and find the average pressure of the atmosphere
in dynes per square centimeter.

101. Characteristics of Fluids. We must now return to the
researches of Pascal on fluids. Both liquids and gases are classed
as FLUIDS, because they both have the
property of offering no permanent resist-
ance to forces that tend to change their
shape. Any portion of a fluid acted on by
forces not equal in all directions flows
freely in the direction in which it is urged
by the greater pressure. Furthermore, all
fluids have perfect elasticity of volume;
that is, if they are compressed ever so
much they immediately resume their

FiQ. 68. Transmission op former volumes when the additional pres-
Fluid Pressure . , *

sure IS removed.
Recognizing these two familiar properties, Pascal reasoned
about fluids somewhat after this manner: Let the bottle. Fig. 68,
be completely filled with any fluid, and let the little circles represent
the elastic particles of the fluid. Suppose the end of the stopper to
have an area of 1 cm*, and let it be pushed in so as to exert a
pressure of 1000 dynes. This pressure of 1000 dynes per square
centimeter will act directly upon the layer of particles adjacent to
it, so that 6, for example, will be pushed downward against 11 and
16, and this pressure will be transmitted without loss through 21,
to the bottom. The pressure transmitted by 6 to 7 and 8 will
tend to push them to the right and left respectively, so that the

FLUIDS 119

pressure is transmitted undiminished to the sides of the bottle.
Furthermore, 8 for example will tend to force 3 upward and 13 down-
ward, thus transmitting the same pressure to those portions of the
top and bottom that lie above and below them. Since all the par-
ticles will be affected in precisely the same manner, it follows that
the pressure will be transmitted not only in the directions men-
tioned, but also in every other possible direction; and therefore the
pressure of 1000 dynes exerted on one square centimeter of the
fluid will cause a pressure of 1000 dynes on every other square
centimeter of resisting surface with which the fluid is in contact.
If the bottle is strong enough to resist this pressure, all parts of the
fluid will be in equilibrium; for if it were not, any portions of the
fluid affected by the unbalanced pressures would move freely in the
directions of the unbalanced pressures until all the pressures were
equalized.

The direction of the pressure on any part of the surface must
be perpendicular to that part; for if the pressure is not perpen-
dicular to that part of the surface upon which it acts, it may be
resolved into two components, one perpendicular to the surface and
the other parallel to it (c/. Art. 53). This parallel component, if it
were exerted, would tend to rotate the bottle about some axis; and
since no such tendency has ever been detected, we believe that no
such parallel component exists. Therefore the pressures are all at
right angles to the surfaces on which they act. It should be care-
fully noted that the truth of this reasoning does not depend in any
way on the shape, size, or number of the particles; and the reader
should avoid the notion that the particles are spherical, or that the
diagram in any way represents their size or number.

102. Pascal's Principle. Having arrived at these conclusions,
Pascal announced them in the following concise statement, which
is known by his name : A pressure exerted upon any part of a fluid
enclosed in a vessel is transmitted undiminished in all directions, and
a^ts with equal force on all surfaces of eqvxil area, in directions per-
pendicular to those surfaces.

A thorough understanding of this principle enables us to ex-
plain all the phenomena of fluid equilibrium.

120

PHYSICS

Fio

69. The Forces are Pro-
portional TO THE Areas

103. Hydraulic Machines. If, for example, we have a vessel
(Fig. 69) consisting of a large and a small cylinder connected by a
pipe and each fitted with a water-tight piston, and if the sectional

areas of these pistons are 1 cm* and
100 cm* respectively, a force of 1 kil-
ogram exerted upon the smaller pis-
ton will produce a pressure of 1
kilogram per square centimeter on
the larger piston, or a total force of
100 kilograms. Thus the force trans-
mitted to any surface by a fluid is
directly proportional to the area of
that surface. "Hence, " said Pascal,
"it follows that we have in a vessel
full of water a new principle of me-
chanics and a new machine for multi-
plyingforces to any degree we choose."
Since the pressures on the two pistons are proportional to their
areas, the mechanical advantage of a hydraulic machine of
this sort is equal to the ratio of the area of the larger piston to that
of the smaller.

With regard to the work done, it should be noted that if in the
example just mentioned the small piston is pushed through a dis-
tance of 1 cm, the large piston will be displaced through only 0.01 cm,
because the liquid forced out of the small cylinder into the larger has
to spread over an area 100 times as great. Therefore, if we mul-
tiply the forces by the corresponding displacements, we find
that the resulting amounts of work are equal. It can easily be
shown that this is true no matter what areas the pistons have, and
no matter what the force and displacement of the smaller piston is;
so that every hydraulic machine conforms to the general law of
machines (c/. Art. 76).

The principle of Pascal is extensively applied in a class of
machines of which the hydraulic press. Fig. 70, is a type. It con-
sists of a large, strong cylinder connected by a pipe with a force
pump (Fig. 71). The piston of the large cylinder has an area many
times larger than that of the pump. The pump piston is worked

FLUIDS

121

by means of a lever and forces water or oil into the large cylinder,
causing the large piston to rise. Thus a bale of cotton, or whatever
substance is to be com-
pressed, is squeezed be-
tween the pressure head of
the large piston and the
heavy frame above it. Hy-
draulic jacks, used for lift-
ing very heavy weights,
work on the same princi-
ple; and hydraulic ele-
vators are operated by the
ordinary pressure in the
city water mains. In this
case, the large piston is
connected with a system
of pulleys by which the
displacement and speed of
the motion are multiplied.
The same principle is ap-
plied in operating drills
and other tools by means
of compressed air.

104. Pressure Due
to the Weight of a
Fluid. Since every fluid
has weight, it follows
that every surface sub-
merged in a fluid in equi-
librium is affected by a
pressure which is due to
that weight alone.

Let us suppose that

/ ^ . Fig. 71. The Hydraulic Press Diagram

a very thin plate havmg

an area of 1 cm^ is placed horizontally in water at a depth of 1 cm.

What is the amount of the pressure on its upper surface?

Fig. 70. The Hydraulic Press

122 PHYSICS

It may be Sjeen from the diagram, Fig. 73, that the water which
rests on this surface is a vertical sided column 1 cm^ in sectional
area; and its volume is 1 cm'. But since the density of water is
1 ^, the mass of this cubic centimeter of water is 1 gm, and
therefore its weight is 980 dynes. Since this mass rests on the
given surface, it must be evident that it exerts a pressure directly

Fig. 72. Mining Coal with a Compressed Air Drill

on it, and that this pressure is nothing more or less than its
weight. If the plate is so thin that its thickness may be neglected,
its under surface will be affected by an equal upward pressure,
because at any given depth the pressure due to the weight of the
overlyjng fluid will be transmitted undiminished in all directions,
as stated in Pascal's principle.

FLUIDS

123

Fig. 73.

Liquid Pressure is Proportional
TO Depth

If the plate were at a
depth of 2 cm, the pres-
sure on it would be 2 gm-
foree or 1960 dynes, be-
cause the weight of the
overlying water column
would be that of a volume
of 2 cm', and so on.

Again, if the liquid,
instea^d of being water,
with a density of 1 ^,
were mercury, which has
a density of 13.6 f^,, the
weight on each square
centimeter would be 13.6
times as great as that of an equal column of water. If the liquid
were alcohol, whose density is 0.8 f^3, the pressure that it would

exert on the submerged surface would
be only 0.8 as great as that exerted by
the weight of water at the same depth;
and so on for other liquids.

Since the pressure due to the weight
of the fluid will be transmitted with un-
diminished force to all equal areas, the
total force on any given surface from
this cause would be directly proportional
to the area of that surface.

Finally, if instead of being horizontal,
the surface were vertical or oblique, the
total force transmitted to it by the
weight of the liquid would be exactly
equal to that which w^ould be exerted
directly on it if it were horizontal, jpro-
vided its center of Thass were at the same
depth; because for every small portion
of the surface having a depth greater than
that of the center, there will be an equal

Fig. 74. Tall Standpipe:
Great Pressure

124 PHYSICS

portion having a depth that is less than that of the center by just the
same amount; and therefore, the mean pressure on every such pair of
small portions of the surface will b(3 equal to the pressure at the
center of mass. The total force on the surface will therefore be
the same as it would be if the surface were horizontal and at the
depth of its center of mass.

Thus the following general statements may be deduced from
Pascal's principle:

The force due to the weight of a liquid in equilibrium, exerted on
any surface submerged in it, is

(1) Directly proportional to the depth of the center of mass of
the surface, the depth being measured vertically from the center of
mass to the level of the free surface of the liquid.

(2) Independent of the direction in which the surface is turned,
provided its center is kept at the same depth.

(3) Directly proportional to the density of the liquid.

(4) Directly proportional to the area of the surface.

These principles enable us to calculate the amount of the force
exerted on any surface by the weight of a liquid in which it is
submerged; we have only to apply this simple rule:

The force du£ to the weight of any liquid, exerted on a surface
submerged in it, is numerically equal to the product obtained
by multiplying together the weight per unit volume of the liquid, the
depth of the submerged surface, and its area,

106. Free Level Surface of a Liquid. We can now understand
why the free surface of a liquid in equilibrium is level, and why,
in a system of communicating vessels, like a teapot and its spout,
the liquid stands at the same level in all the vessels. For if the surface
were higher at one place than at any other place, the liquid pressure
there would be greater, and the liquid therefore would flow from the
place of higher level to the places of lower level, until no part of the
liquid were higher than any other.

106. Oases. Returning now to gases, we can appreciate that
all we have said of liquids applies equally well to gases, with two
important exceptions. First, the pressure in a gas is not propor-

FLUIDS 125

tional to the depth, because gases are easily compressed, so that
the lower portions are denser than the upper ones. Second, gases
have no such thing as a free level surface, for they tend to expand
indefinitely, so as to fill completely the vessels in which they are
confined.

107. The Air Pump. The phenomena of atmospheric pressure
were very thoroughly investigated by Otto von Guericke (1602-
1686), am eminent engineer, and burgomaster of Magdeburg.

Guericke began experiments on the vacuum by filling a cask
with water, and then pumping the water out of an opening at the
bottom. Finding this method very imperfect, he finally suc-
ceeded in inventing a fairly efficient air pump, which, a few years
later, was much improved by Boyle and Hooke, in England.

108. The Magdeburg Hemispheres. Guericke made many
experiments, one of the cleverest of which was that of pumping the
air out of a pair of hollow iron hemispheres having smooth rims
which fitted very accurately together. When the air was pumped
out of these, it was found, as Guericke expected, that great force
must be exerted in order to separate them; because the pressure due
to the weight of the air above them held them together. Since the
force required to overcome this excess of external pressure is con-
stant, no matter in what direction the axis of the hemispheres is
turned, the experiment proves that at any given place the pressure of
the atmosphere acts with equal force in all directions.

This experiment was made by Guericke in the presence of
Emperor Ferdinand II and the Reichstag, with hemispheres 1.2
feet in diameter. The force of sixteen horses was required to sepa-
rate them. Of course eight horses would have done as well if he
had attached one of the hemispheres to a wall or post, but the
dramatic effect might not have been so great. Plate V is a photo-
graph of the picture that appeared in Guericke's book.

109. Density of Air. Guericke was the first to demonstrate
that air has weight by pumping some out of a hollow globe instead
of forcing it in as Galileo did. He used a vertical tube of water

126 PHYSICS

as a barometer or vacuum gauge. The density of air has been
very accurately determined by weighing in accordance with the
method of Guericke, and is .001293 ^3, at 0° Centigrade and
76 cm barometer pressure.

Since we have learned that the pressure of the Jitmosphere is
about one kilogram-force on each square centimeter, the question
naturally arises as to how we are able to withstand so great a pressure
on our bodies. The reason is that our blood and tissue cells
contain air at the same pressure. The presence of this air can be
demonstrated in an experiment with the "hand glass." This is a
receiver which fits on the plate of the air pump, and has an opening
at the top to which the palm of the hand can be fitted air tight.
When the air is pumped out of the receiver, the hand is not only
pushed down with great force by the weight of the overlying air,
but also the fleshy part of the palm swells out and extends
through the opening into the receiver. This is because the pressure
of the outside atmosphere has been removed from that part of the
hand, and the air within the hand, being now freed from this pressure,
expands and distends the cells in which it is confined. It is for this
reason that aeronauts and mountain climbers often suffer great in-
convenience, for, as they ascend, the pressure of the atmosphere
diminishes so rapidly that the blood is forced to the surface by the
pressure of the air within their tissues. This unbalanced pressure
puts an unwonted strain on the blood vessels, which often causes
some of them to burst.

110. Theory of Pumps. We are now in possession of all
the information needed for explaining why a pump acts, as
well as how it acts. When the piston P; Fig. 75, is with-
drawn, it removes the atmospheric pressure from the fluid in the
cylinder C; and the fluid is pushed into the suction pipe by the
atmospheric pressure outside. Therefore the resultant force tend-
ing to push the fluid into the cylinder is equal to the atmospheric
pressure diminished by the weight of the fluids in the suction pipe.
If the atmospheric pressure exceeds this weight, the inlet valve iv
will be pushed open, and some of the fluid will be pushed from the
suction pipe into the cylinder. Meanwhile the pressure due to the

FLUIDS

127

weight of the fluid in the outlet pipe, plus the atmospheric pres-
sure, keeps the outlet valve ov closed, and prevents the reentrance
of any fluid from the outlet pipe. When
the piston is pushed in, it exerts on the
fluid m the cylinder a pressure which closes
the inlet valve, opens the outlet valve, and
pushes the fluid out through it.

Thus every double stroke removes a
volume of fluid which is very nearly equal
to the area of the piston multiplied by the
length of the stroke. If the fluid to be
pumped is air or any other gas instead of
to be pumped into or out of a closed

Fia. 75. Pump Diagram

a liquid, and if it is
vessel, the pressure in the
closed vessel is mainly caused, not by the weight of the confined
gas, but by its elasticity.

111. Archimedes's Principle. Another important corollary
deducible from Pascal's principle, is known by the name of Archi-
medes. Let Fig. 76
represent a vessel filled
with liquid to the level
ab, in which is sub-
merged a rectangular
solid cdef, and let us
find the resultant force
due to the weight of
the liquid and tending
to move the solid.

The resultant of all
the horizontal forces is
zero, because every
such force is opposed
by an equal and oppo-
site force at the same
depth on the opposite side. But how about the vertical forces?
The force on cd is equal to the weight of a column of liquid repre-
sented by cdhg, acting downward; and the force 'on fe is equal

Fig. 76. Buoyant Force Equals Weight
OF Water Displaced

128 / PHYSICS

to the weight of the column of liquid jehg, transmitted as described
by Pascal's principle, and acting upward. The resultant of these
two forces, and hence of all the forces, is equal to their arithmetical
difference, which evidently is equal to the weight of a volume of the
liquid represented by Jecd, This volume is that of the body, and
therefore of the liquid displaced by it.

Archimedes of Syracuse (287? — 212 B.C.), who was the greatest
natural philosopher that lived before the time of Galileo, must have
known much of what we have learned about the equilibrium of
liquids, for he discovered the principle that we have just reached
and announced it substantially as follows :

A body immersed in a fluid is huoyed up by a force that is equal
to the weight of the fluid displaced by it.

The foregoing argument for the principle of Archimedes does
not depend for its conclusion on the kind of fluid, npr on the
depth to which the body is submerged. It has also been shown to
apply to bodies of all shapes and sizes.

112. Floating Bodies. By Archimedes's principle, we can
easily predict whether a body will float or sink in any fluid, whether
that fluid be a Hquid or a gas. Thus, if the weight of the body is
greater than that of an equal volume of the fluid, the body will sink
to the bottom, of the fluid; if the weight of the body is less than that
of an 'equal volume of the fluid, the body, if submerged, will float
upward; if the weight of the body is exactly equal to that of an equal
volume of the fluid, the body will remain wherever it is placed within
the fluid.

From the foregoing principles it follows that if a body which is
lighter than the same volume of a given fluid be placed in that fluid,
it will rise or sink (depending on where it is placed), and will
come to equilibrium when it displaces a volume of the fluid that
weighs exactly as much as it does. The buoyant force will then
exactly balance its weight. This special case of the application
of the principle of Archimedes is known as the principle of flo-
tation.

The principle of Archimedes describes implicitly the behavior
of a boat or a balloon. The more heavily a boat is loaded, the

FLUIDS

129

deeper it will sink. Why? Its gross displacement is the weight

of the maximum volume of the water that it can safely displace.

The weight of its maximum safe load is evidently the difference

between its weight and its gross

displacement. A very great load

necessitates a correspondingly

large displacement, which tends

to diminish the speed that it can

attain.

Boats are made practically un-
sinkable by reserving a sufficient
amount of the interior space for
separate water-tight compartments,
so that the boat can not sink by
taking in water unless a number of
the compartments are punctured
at the same time. Submarine
boats are made to sink by letting
water into their compartments, and
are made to rise by forcing it out
with strong pumps.

The load that a balloon, Fig.
77, can support is equal to the
weight of air displaced, dimin-
ished by the sum of the weights of
the balloon, car, rigging, con-
tained gas, and ballast. If the aeronaut wishes to ascend, he
diminishes the gross weight of the balloon by throwing out ballast.
If he wishes to descend, he diminishes the volume of the balloon
by opening a valve at the top and letting some gas escape. This
diminishes the buoyant force. Why?

Fig. 77. Santos Dumont's
"Runabout"

113. Determination of Density. Among the important appli-
cations of the principle of Archimedes are several methods of de-
termining density. The following example illub'trates one of these.
A piece of rock weighs 25 gm. When suspended from the balance
pan so as to be wholly submerged in pure water, it is balanced by

130

PHYSICS

15 gm. The buoyant force of the water on it is equal to its appa-
rent loss in weight, or 10 gm. We have learned in Art. 32 that
1 cm' of water has a mass of 1 gm; we know, therefore, that the
volume of the 10 gm of water is 10 cm'. Since the volume of the
rock is the same as that of the displaced water, or 10 cm', and
since the mass of the rock was found to be 25 gm, its density is

mass __ z5 gm _ « ^ gm
volume 10 cm' * cm'*

Archimedes used this method in determining for Hiero, king
of Syracuse, whether some gold, which he had furnished to an
artisan to be made into a crown, had been partly
retained by that worthy, and the weight made up
by alloying with baser metal.

When we are possessed of the knowledge
gained by the experimental researches of Galileo,
Torricelli, Pascal, and their eminent contempo-
: raries, it seems easy enough for us to under-
stand and apply the principle of Archimedes;
but when we remember that Archimedes lived
nineteen centuries before these men, we can not
> but marvel at the genius which enabled this
great philosopher to think so clearly about this
principle and its applications to liquids.

FiQ. 78
Boyle's Tube

114. Boyle's Experiments. Among those
scientists of the seventeenth century who con-
tributed so largely to our knowledge of fluids
was Robert Boyle of England (1627-1691). His
greatest discovery, that of the law that goes by his
name, was made during an investigation upon the elasticity of air.
In order to find out what elastic force compressed air was able
to exert, and what would be the effect of increased external pressures
on the corresponding volumes, Boyle provided a tube. Fig. 78,
"which, by a dexterous hand and the help of a lamp, was in such
manner crooked at the bottom that the part turned up was almost
parallel to the rest of the tube." The shorter leg was closed and
the longer open.

FLUIDS 131

He started with the column of confined air 12 inches long, and
with the mercury at the same level in both legs of the tube. Since
the columns of mercury in the two legs then balanced each other^
the pressure on the confined air was simply that of the atmosphere.
(Why?) Reading his barometer, he found this atmospheric pressure
equal to 29 inches of mercury. More mercury was then poured
in, till the column A was 29 inches long; the pressure on the
confined air was therefore twice 29 inches. (Why?) When the
confined air column was under this pressure, its length e was found
to be 6 inches. Thus, when the pressure had been doubled, the
volume was reduced to one-half. Another 29 inches of mercury
reduced the volume to 4 inches, or one-third; and so on.

115, Boyle's Law. As a result of extended experiments,
therefore, Boyle announced the following law: The volume of a
given mass of gas, at a constant temperature, varies inversely as
the pressure that it supports.

In symbols, if V and F' represent any two volumes of a certain •
mass of gas at a constant temperature, and P and P' the corre-
sponding pressures, Boyle's law is represented by the equation

V P'

f, = ^orrP = F'P'.

The latter form of the equation shows that the products obtained by
multiplying together each volume and its corresponding pressure
are all equal. Therefore, calling this constant product K, we may
represent Boyle's law by the equation VP = K and express it in
ordinary language as follows:

At a constant temperature, the product of the numbers represent-
ing the volume and pressure of a given mass of any gas is a constant
quantity , This law has been verified by a great many experiments,
and is found to be approximately true for all gases within certain
^ide limits; but at certain temperatures and pressures for each gas
the law fails, notably when, on account of great compression, or
low temperature, or both, the gas is about to liquefy.

Aeriform bodies, like air or steam, are classified as perfect
GASES when they act in accordance with Boyle's law, and as
VAPORS when they do not so act.

132 PHYSICS

The graphical representation of Boyle's law is very interesting,
and of great assistance in the solution of problems connected with
the clastic pressures of gases in the cylinders of engines using the
energy of steam, gas, or compressed air.

SUMMARY

1. The average pressure, at sea level, of the atmosphere bal-
ances the weight of a column of mercury 76 cm high. This pres-
sure is equal to 1032.84 grams-weight per square centimeter, and
is called one atmosphere.

2. The barometer is used to measure atmospheric pressure.
It is applied: 1, in weather observations; 2, in determining eleva-
tions; 3, in many experiments with gases.

3. A pressure exerted on any portion of a fluid enclosed in a
vessel is transmitted undiminished in all directions, and acts with
equal force on all surfaces of equal area, in directions perpendicular
to those surfaces. (Pascal's principle.)

4. Pressure due to the weight of a liquid in equilibrium is pro-
portional to its depth and to its density.

5. The pressure due to the weight of a gas is not proportional
to its depth. •

6. Liquids have a free level surface; gases do not.

7. Gases tend to expand indefinitely.

8. A body immersed in a fluid is buoyed up by a force equal
to the weight of the fluid displaced. (Archimedes's principle.)

9. When a body is submerged in water, the number of gm of
water displaced by it is equal to the number of cm' in its volume.

10. The volume of a given mass of any gas at constant tempera-
ture is inversely proportional to the pressure that it supports.
(Boyle's law.)

QUESTIONS

1. Describe the simple mercurial barometer, as arranged in the
experiment of Torricelli.

2. Why is the mercury column thus upheld shorter than the column
of water in the suction pipe of a pump?

3. Is there any air in the space above the mercury in a Torricellian
tube?

FLUIDS 133

4. Explain briefly why a falling barometer indicates stormy weather,
and a rising barometer, fair weather.

5. Explain how a barometer may be used for measuring altitudes
above sea level.

6. Why is it that if a balloon were ascending imiformly a barometer
column would fall faster near the surface of the earth than it would at
a greater altitude?

7. Show how a mechanical advantage may be obtained from a body
of fluid, as in the case of a hydraulic press. Show that a hydraulic
machine conforms to the general law of machines.

8. State four facts about the force due to the weight of, a liquid in
equilibrium, and show how they are deducible from Pascal's principle.

9. Suggest some experiments by means of which these fo\ir facts
might be verified.

10. State a rule, derived from these facts, for determining the
total force exerted by the weight of a liquid on a surface submerged in
it. In this rule, how is the depth to be measured?

11. By means of a suitable diagram, explain why the free surface
of a liquid in equilibrium is level.

12. Describe the experiment of the Magdeburg hemispheres. Tell
what it proves, and how it proves it.

13. Describe Guericke's method of proving that air has weight,
and contrast it with Galileo's.

14. Explain why animals can withstand the great crushing force
of the atmosphere.

15. Draw a sectional diagram of a force pump, and fully explain
both how and why the fluid is propelled through it.

16. Discuss the application of the principle of Archimedes to a
boat, and to a balloon.

17. Explain how to calculate the load that a given boat or balloon
can support.

PROBLEMS

1. On a mountain the barometer reads 45 cm; what is the pressure
of the atmosphere there (a) in ^^^ (6) in ^|5?

2. When the atmospheric pressure supports a column of mercury
75 cm high, how high a column of water will it support? How high a
column of alcohol? Take the densities, as: mercury, 13.6; water, 1;
alcohol, 0. 8.

3. A plunger whose cross-sectional area is 4 cm^, is pushed into a
cylinder full of oil with a force of 5 X 10* dynes; what pressure in -^I^
must be sustained by the walls of the cylinder? If the end of the cyl-
inder has an area of 300 cm^, what is the total force exerted on it?

134 PHYSICS

4. The pump plunger A, of a hydraulic press, Fig. 71, has
an area of 5 cm^ and the ram B an area of 1000 cm^; what is tlie
mechanical advantage? How many kilograms-force must be applied
to the plunger in order that the pressure head of B shall exert a total
force of 9 X 10* Kg-force?

5. When a man presses down on the lever of the press of problem
4 at a distance of 60 cm from the fulcrum, what is the mechanical ad-
vantage of the lever if the plunger is 10 cm from the fulcrum? What
force must the man exert in order that the force on the plunger may .
be 450 Kg-force? What is the total mechanical advantage of the press
when the lever is used?

6. When the standpipe in Fig. 74 contains water to the height
of 30 m, what pressure in ?ni^°p5 does the water exert at the bottom?
If one of the steel plates on the side there is 3 m long and 1.8 m high,
what total force must it withstand?

7. Find the pressures in in^_2r£? ©n the sides of the standpipe
at depths of 1 m, 2 m, 5 m, 10 m, 20 m, 25 m, and, using any con-
venient scale, plot a graph showing the relations of pressure to depth
of the water. Does this graph suggest to you anything about the
relative thicknesses that the steel plates' must have at these various
depths in order not to burst?

8. Fig. 79 represents an experiment devised by Pascal to verify
the conclusions stated in Art. 104. The bottom is held on to each of

the three vessels in turn by the pull of the
cord only. If the distance from the pointer
to the bottom is 10 cm, and the area of
the bottom is 50 cm^, how many gm-force
on the other balance pan are required to
hold the bottom on when the cylindrical
vessel is filled up to the pointer? How
many for the wide-topped vessel? How
many for the narrow-topped vessel?
Would the result be the same if a vessel
Fig. 79 having any other shape were used, pro-

vided the other conditions remained the

same? Explain how this apparatus may be used to verify statements

1, 3, and 4 of Art. 104.

9. Taking 1 -^^^^® as the pressure of the atmosphere and 36

cm as the diameter of the Guericke hemispheres, Plate V, calculate
the force with which they were held together, assuming that he got
a perfect vacuum inside them. N. B. — The surface to be used is the
area of greatest cross-section, not the surface of the sphere. (Why?)
The latter would give the crushing force; calculate it.

FLUIDS

135

10. A balloon contains 300 m^ of illuminating gas, which weighs
One m'of air weighs 1.3 Kg; what weight, including its own,

will the balloon support?

11. A canoe weighs 75 lb., 1 ft' of water weighs 62.4 lb. How many
ft^ of water must the boat displace when it is carrying two persons,
weighing together 240 lb.?

12. The weight of a steamer is 6000 tons, and its gross displace-
ment is 10,000 tons; what load can it carry?

13. Fig. 80 represents a syphon. Suppose the atmospheric pressure

is 103

gm-force
cm2

and that EA == 10 cm, DB = 20 cm. With how many

At B?

What is the result-
What will

^™cm2^^ ^^^^ ^^® water press down at A ?

ant pressure in the direction ACB? In the direction BCA?

the water do? Would the syphon work on a moun-
tain top? In a vacuum? Over how great a height

can water be raised by it when the barometer

stands at 75 cm?

14. Fig. 81, the cylinder C is hollow and has a

capacity of 100 cm^. P exactly fits it. P and C

are balanced, as shown, but without any water in

the vessel. The water is then placed in the vessel

under P; will it remain submerged? If water is

poured into C until equilibrium is restored, how

many gm will be required? How many cm'?

15. Given: the mass of a
piece of glass = 50 gm, the
weight of the glass in water
= 30 gm, the weight of the
glass in gasoline = 36.26 gm.
Required : the volume of the
glass, its density, the mass of
gasoline that has the same vol-
ume as the glass, and the den-
sity of the gasoline.

16. A piece of wood having a mass of 37.5 gm is attached to a
piece of lead whose mass is 166.5 gm. The weight of both, when sub-
merged in water, is 139 gm. The lead alone weighs in water 151.5
gm. Find: (a) the volume of water displaced by both together; (b) the
volume of the lead; (c) the volume of the wood; (d) the density of the
wood; (e) the density of the lead.

17. If you hold your finger at the outlet valve of a bicycle pump
when the pjston is at the top of the cylinder, and if the piston is then
pushed half-way down, what is the pressure on your finger in 5l:^^£®

if the pressure gf the atmosphere is 1 ^j? What are the pressures when

Fig. 80

Fig. 81

Lengths

Pressures

29.1

35.3

39.3

44.2

50 3

58.8

70.7

87.9

3.2

107.8

136 PHYSICS

the piston is i, A» J o^ *he way down? If you pump up your tire until
the pressure in it is doubled, by how much is the density of the air
in it changed?

18. Appended are some of the data
obtained by Boyle in his experiment
(Art. 114) : Choosing a convenient scale,
plot the lengths of the air columns as ab-
scissas, and the corresponding pressures
as ordinates. The graph will then rep-
resent the relation PV = const. What
does the graph show about the pressure
when the volume becomes very large?
What about the volume when the pres-
sure becomes very large?

SUGGESTIONS TO STUDENTS

1. By means of a rubber tube connect a bubble pipe with the gas
fixture, let the gas blow soap bubbles for you, and see what they will
do. Can you explain their behavior?

2. Visit the water works and find out all you can about the pump-
ing engines and the pumps. How is the pressure regulated by means
of air chambers connected with the inlet and outlet pipes? Find out
whether there is a standpipe connected with the works, and if so,
what its use is.

3. Can you explain how your bicycle pump works? Find a com-
pressed air tool at work (a riveter on a steel framed building or bridge,
or a cutting tool at a marble works), and learn what you can about
how they work.

4. When a sail-boat is tipped from the position in which the deck
is horizontal, what moment of force does most of the tipping? What
moment tends to restore it? Investigate this interesting and important
problem of the stability of a boat and find out what you can about it.

5. Consult a book on physiology in which the action of the heart
is described. Can you understand wherein it resembles a force pump?
In what essential detail does its action dififer from that of an ordinary
pump?

6. If you are interested in the air ship shown in Fig. 77, read Santos
Dumont's book on My Air Ships (N. Y. Century Co., 1904)."

CHAPTER VII
HEAT

116. Heat and Work. In the preceding chapters we have
studied the operation of the locomotive and of other machines and
learned of motions and mechanical efficiencies. How is it with
steam engines and gas engines? Every one is familiar with the
fact that every engine consumes fuel in some form, and that
without the fuel it will not move at all. Hence, all engines
must in some way derive the energy with which they do work
from the fuel that they bum, i.e., an engine is simply a device
for converting heat energy into mechanical work.

The questions that arise in connection with the conversion of
heat into mechanical work are many and interesting. Thus,
how can we measure heat? How do bodies change when they
are heated or cooled? What of the process of converting water
into steam? Is heat absorbed in this operation? Does steam act
as a gas and obey Boyle's law, or does it act differently? How
is the heat transferred from the fire to the water in the boiler
and from the water into steam? Is there any definite relation
between heat and mechanical work? Is the efficiency of a steam
engine high or low, and how is it determined? On what factors
does the efficiency of such an engine chiefly depend?

117. Heat Sensations Xrnreliable. Perhaps the most familiar fact
about heat is that some things feel hot to our touch while others

,feel cold. Yet our ability to judge how hot a body is depends on
a number of varying circumstances, and at best is limited. For
example, if we take three basins of water, one hot, one lukewarm
and one cold, and place the right hand in the hot water and the
left in the cold, and then transfer both hands to the lukewarm water,
this latter will seem cold to the right hand and hot to the left.
Hence, we can not rely on our sense of touch for accurate informa-

137

138

PHYSICS

tion concerning differences of temperature. What, then, may we
use?

118. Galileo's Thermometer. The first to give any scientific
answer to this question was Galileo. He blew a bulb on the end
of a glass tube of small bore and after slightly warming the bulb
placed the end of the tube in a vessel of colored water .(Fig. 82).
When he warmed the bulb with his hand the liquid
in the tube moved downward, showing that the air
in the bulb expanded. Also conversely, when he
cooled the bulb the liquid in the tube moved upward,
showing that the air in the bulb contracted. He thus
showed that air expands when it is warmed and con-
tracts when it is cooled; and he suggested the use of
this property of air for detecting small differences of
temperature.

It is probably not necessary to state that Galileo's
suggestion has been universally adopted for scientific
work, although the instrument which he devised is
practically useless as a thermometer, because the liquid
in it is exposed to the pressure of the atmosphere. Since,
as we learned in the last chapter, this pressure is
always changing, the small column of liquid moves
Fig. 82 when the atmospheric pressure changes as well as when
Thermom- the temperature changes. Therefore, we can not be
sure that a given position of the liquid indicates the
same temperature at different times. Fortunately, allowance can
be made for the error thus introduced, provided the barometer
is observed at the same time with the thermometer, so that the
pressure on the air in the bulb is known. But even so, how may
we determine the amount of the change in temperature correspond-
ing to a given motion of the liquid in the tube?

119. The Temperature Scale. We can not answer this question
until we have adopted a scale of temperature, and defined the units
in terms of which we shall measure differences of temperature. In
order to establish such a scale it is necessary to have some fixed

HEAT 139

temperature which may be used aS the zero from which to count,
and also to have some unit difference of temperature. It has been
found convenient to adopt the temperature at which ice melts as
the ZERO temperature; hence, in scientific work this is called
a temperature of zero degrees.

In order to determine a unit ' difference of temperature, we
•must select some other fixed temperature, and then define the
interval between the zero and this other temperature as a certain
number of degrees. The second fixed temperature that has been
adopted by scientists is that of water boiling at normal barometer
pressure, i.e., 76 cm. The interval between the temperature of
melting ice and that of boiling water has been divided into a hun-
dred equal^ temperature intervals called degrees. Another temper-
ature scale, called Fahrenheit's, is in common use, but the one just
defined is generally used in scientific work. Since, in defining
this scale, the fundamental temperature interval is di>dded into
one hundred equal parts called degrees, it is called the centi-
grade SCALE. Temperature degrees in this scale are denoted by
the symbol °C. For example, 40° C. means forty degrees of the
Centigrade scale. As the temperature does not involve either gm,
cm, or sec, it has no symbol in the terms of these units.

Having defined our temperature units, we are now in a position
to put a scale on our thermometer. This is done by placing the
instrument in melting ice, marking the position of the drop of
liquid in the tub», then placing the instrument in the steam over
boiling water, and marking the position of the drop of liquid when
it has become stationary. The interval on the tube between
these two marks is now to be divided into one hundred parts
representing equal temperature intervals.

120. Change of Volume at Constant Pressure. Let us now
ask what the relation is between the volume of the air in the bulb
and an" increase in temperature of 1°. This relation has been deter-
mined experimentally with great accuracy, and it has been found
that when a given mass of gas is heated from 0° to 1° its volume
increases ^, when heated from 0° to 2° its volume increases gfg,
of its volume at 0°, etc., i.e., for every change of 1° in tem^perature,

140 PHYSICS

the corresponding change in volume is 273 of the volume at 0°.
This ratio has been found to be the same for all gases and for all
changes of 1° in temperature. It is called the coefficient of
EXPANSION OF GASES. Since the measurements by which these
facts were first established were made by Charles and Gay Lussac,
this relation is known as the law or Charles and Gay
Lussac.

If we let V represent the volume of the gas at any temperature
/, and Vq its volume at 0° C, then, since the final volume (V) is
equal to the volume at 0° (Vq), plus the increase in volume (273 ^o)>
this law is expressed analytically as follows:

^=^«0 + 2f3>

1 . ' V

On factoring out ^;^, this equation becomes V = — -^ (273 + i).

V
Since r~ is a constant for any mass of gas, we see that at constant

pressure the volume of a given mass of gas is proportional to
(273 + t).

It is to be noted that this equation accounts for changes of
volume due to changes of temperature only; and hence, in stating
this equation it is assumed that the pressure of the gas remains
constant.

121. Change of Pressure at Constant Volume. We have
just found how the volume changes when a gas is heated at constant
pressure; let us now try to find out how the pressure varies when
the gas is heated while its volume is kept constant.

In order to do this we must add to the thermometer of Galileo
a device for governing and measuring the pressure in the bulb
This device usually consists of a rubber tube K, Fig. 83, which is
fastened at one end R to the glass tube of the thermometer and
at the other R to another similar piece of glass tubing. This
rubber tube is then filled with mercury until the mercury appears
above both its ends. The instrument as thus arranged is called
an AIR THERMOMETER. It will be readily seen that by raising or

HEAT

141

lowering the free end ii' of the rubber tube we can increase or
diminish the pressure of the air in the glass bulb.

If the glass bulb is now heated, the air within it will expand
and depress the mercury at the end R of the rubber tube. To
compress the air to its original volume, the other end of that tube
must be elevated until the mer-
cury in the thermometer re-
turns to its former level d.
Thus, the increase in pressure
produced by the rise in temper-
ature is balanced by the pressure
due to the weight of the mercury
in column h; and the total pres-
sure may be found by adding
that of the column h to that of
the atmosphere as read from
the barometer (c/. Art. 115).
What is the relation between
the increase in temperature
and that in pressure when the
volume of gas is kept constant?

We can find the answer to
this question by measuring the
changes in temperature and the
corresponding changes in pres-
sure. This has been carefully
done, and the relation is found
to be similar to that between
change of volume and change
of temperature at constant pres-
sure. It is stated as follows : when a given mass of gas is heated

at constant volume, the pressure increases - — of the pressure at 0°

for every change of 1° in temperature. This is true also for all
gases and for all their changes of temperature.

If we let P represent the pressure of a given mass of gas at any
temperature t, and Pq its pressure at 0° C, then, since the final

Fig. 83. Air Thermometer

142 PHYSICS

pressure (P) is equal to the pressure at 0° (Pq), plus the increase
in pressure ( ^— - P^Y the result is expressed analytically as follows:

'=^»0+2f3)"

Or, factoring out the 2^3' ^ = ^^ (^73 + 0-

We can therefore find the value of t with the air thermometer
by subi^titutirig the observed values of P and P^ in this equation,

273P
and solving it for /. Thus t =— ^ 273.

P
Since -^—^ is a constant for a given mass of gas, we see that

at constant volume the pressure of a given mass of gas is propor-
tional to (273 -\- t). It will aid the memory to note that the equa-
tions of this article, and those of Art. 120 are similar in form.

The air thermometer is not a simple instrument to handle,
therefore temperatures are generally measured by the ordinary
mercury thermometer, but it must not be forgotten that the air
thermometer is the standard to which the mercury tJiermometers are
all referred.

122. When Pressure, Volume, and Temperature Change.
Now, it has been shown in Art. 120 that, at constant pressure,
273 + t is proportional to the volume V. It now appears that at
constant volume this quantity is also proportional to the pressure P.
Therefore it follows that 273 + t is proportional to the product of
,VandP; i.e., PV = Constant X (273 + t). The numerical value
of this constant depends on the density of the gas and the units
in which the quantities are expressed. For solving most gas prob-
lems we may express this relation in a more convenient form.
Thus if V is the volume of a mass of any gas at <° C. and pressure
P, and F' its volume at temperature f° and pressure P', then

VP _ 273 + <
V'P' 273 + f ^^

This equation expresses the relations of pressure, volume, and
temperature for gases. It means that the product of the volume and

HEAT 143

pressure of a given mass of gas is directly proportional to 273 + its
temperature.

123. Absolute Temperature. Since the quantity PV is not
proportional to the temperature as measured on the Centigrade
scale, but is proportional to 273 plus that temperature, it is convenient
for work of this kind to conceive that the zero temperature is placed
at —273°. This new zero of temperature is called the absolute
ZERO, and temperatures measured from it are called absolute
temperatures.

Therefore 273 + t represents the absolute temperature.

124. Expansion of Solids and Liquids. Thus far we have
studied the changes in gases caused by changes in their tempera-
tures. Do solids and liquids expand when they are heated, and
contract when they are cooled? Everybody knows that they do.
Just as in the case of gases, the volume of every solid and liquid
is changed by a certain fraction of itself for every degree that the
temperature changes. This fraction is called the coefficient of
cubical expansion. Unlike gases, however, each liquid and
solid has its own coefficient of expansion which is characteristic
of it.

If V represent the volume of any solid or liquid at a tempera-
ture t, Vq its volume at 0° C, and c its coefficient of expansion,
then F = Fo (1 + ct) (cf. Art. 120).

Similarly for solids when the change in length only is important,
the fractional change in length for 1° C. is called the coefficient
OF linear expansion. If a represent this fraction, L the length
of the solid at any temperature /, and Lq the length at 0° C, then
i = Zo (1 + at).

These coefficients are needed for the solution of many impor-
tant problems which arise in everyday life because of the expansion
and contraction due to changes in temperature. For example,
when railroads are constructed in winter, spaces must be left
between the ends of the rails to allow for the expansion in summer.
Long span bridges must have their ends placed on rollers; and
provision must be made in a hot water heating system for the
expansion of the water and of the pipes.

144 PHYSICS

125. Measurement of Heat. We have defined units in
which we may measure temperature, but does the determination
of the temperatures give us any information about the amount
of heat? May not a small body and a large body have the same
temperature but contain very different amounts of heat? Hence,
in order to answer the question as to the quantity of heat absorbed
by an engine or by any other device or body, we must make a
further definition as to the units in which this heat is to be meas-
ured. The unit universally adopted by physicists for quantities of
heat is the quantity of heat absorbed by one gram of water when
heated from 15° to 16° C This unit is called the gram calorie.
Thus, if one liter (1000 cm) of water is heated until its temper-
ature has risen through 100°, the quantity of heat thus imparted
to it is 1000 X 100 = 100,000 gram calories.

126. Specific Heat. But does it require equal amounts of heat to
increase the temperature of all substances by 1°? Evidently not,
for it is well known that it requires more heat to raise the tempera-
ture of a gram of water through one degree than to raise the same
mass of any other substance through the same temperature interval.
In order to compare the heat capacities of different substances,
it is, therefore, convenient to express their ability to absorb heat
in terras of the heat-absorbing power of water. We may call this
heat-absorbing power Specific Heat, and define it as the ratio

heat absorbed by 1 gm of the given substance in warming 1°
heat absorbed by 1 gm of water in warming 1°
i.e., the specific heat of any substance is the number of calories
required to warm 1 gram of it through 1° C

The numerical value of the specific heat of any substance may
be determined by experiment in a number of different ways. One
of the simplest of these is illustrated by the following example:
100 gm of aluminum clippings at 98° C. are stirred into 200 gm
of water at 2° C; and the mixture comes to the temperature of
11.5°. If h represent the specific heat of aluminum, the heat
given up by it in cooling to 11.5° is

hX 100 gm X (98°- 11.5°).
Sp. Ht. X mass X change of temperature.

HEAT 145

The heat absorbed by the water in warming to 11.5° is
1 X 200 gm X (11.5° - 2°).
Sp. Ht. X mass X change of temperature.

The heat absorbed by the water must be equal to that given

out by the aluminum; so that we may form the equation,

A X 100 X (98 - 11.5) = 1 X 200 X (11.5 - 2), or A = 0.225.

In performing the experiment care must be taken to let as
little heat as possible enter or escape, since it is assumed by the
equation that none does so; and proper allowance must, in general,
be made for the heat absorbed or emitted by the vessel that holds
the w^ter. Vessels used for this purpose are called calorimeters,
and are usually made of metal, and surrounded by a box designed so
as to prevent heat from entering or escaping. *

127. Steam. One other important phenomenon connected
with an engine must be understood before we can determine its
efficiency. This is the phenomenon of making steam. Heat
is absorbed in this process; for a boiling kettle apparently
ceases to emit steam shortly after being removed from the fire; and
the greater the surface exposed to the air, the faster the water
cools. But does water at any temperature ever cease emitting
steam? If not, why do we have to heat it so hot in order to make
it produce steam for use in the engine? Does water boil always
at the same temperature, and what is the nature of the phenom-
enon we call boiling? In order to ilnd answers to these questions,
consider first a glass fruit jar of water exposed to the air. If
this jar is placed in a warm room, v/hat will happen to the water?
Suppose that the cover is sealed on to the jar, what will then happen
to the water? Will it evaporate at all? If so, how much? If
not, why not? Suppose that we place the open jar under the
receiver of an air pump and exhaust the air, will the water then
evaporate; and, if so, to what extent? Will it evaporate in the
same room faster if it is hot than if it is cold?

128. Evaporation. It is well known that water evaporates
when left in open dishes, and that in the same room the evaporation

146 PHYSICS

is faster the hotter the water; hence, we will all agree that water
passes into aqueous vapor or steam at all ordinary temperatures.
We may also grant that the reason why water evaporates from an
open dish but does not disappear from a closed jar, is that the room
is so much larger than the jar, and hence is capable of holding
very much more water vapor than the space in the jar can hold.
But how much water vapor will a given volume hold, and is this
amount the same at all temperatures?

Elaborate experiments were necessary to determine these points,
and the results show that water will always evaporate until its
vapor exerts a certain pressure on the walls of the vessel con-
taining it, and that this pressure can not, at a given tempera-
ture, exceed a certain definite value. When the pressure reaches
this maximum value the space in the vessel is said to be
SATURATED, and hence this maximum pressure is called the pres-
sure of the saturated vapor at the given temperature. This pres-
sure is independent of the other contents of the vessel, and
depends only on the temperature of the water and its vapor. The
case is similar for other liquids. Hence, we see that evaporation
takes plaice wherever there is an exposed surface of liquid, and
continues until the vapor attains this maximum pressure of satu-
ration.

Further, as long as any liquid remains in the closed space we
can not increase the pressure of this saturated vapor, provided
the temperature remains constant. Nor can we alter that pressure
by changing the size of the space: for if we increase the space, more
vapor is formed; if we decrease it, some vapor is condensed into
liquid; and the pressure exerted by the vapor remains constant.
Hence, we may say in general that the pressure exerted at a given tem-
perature by saturated vapor in contact with its liquid is always the
same. Of course, the pressure exerted by the saturated vapors
of different liquids at a given temperature are not necessarily
the same.

A simple experiment will make these matters clear. In the
barometer tubes 6, 6', 6", Fig. 84, the three mercury columns at
first stand at the same height, depending on the amount of the
atmospheric pressure at the time. Now, with a medicine dropper,

HEAT

147

^11 1

ti ii 'j

1 '■ ■''
1

insert a few drops of water under the end of the tube b\ tap
the tube gently, if necessary, till the water rises to the top of
the mercury column, and observe the result. The
water evaiporates till the water vapor exerts its
pressure of saturation corresponding to the temper-
ature of the experiment. Manifestly, this pressure
is measured by the depression ct of the niercury
column. Similarly, a few drops of ether, inserted
in 6", partially evaporate, and exert the pressure
of saturated ether vapor corresponding to this
temperature. This pressure is measured by the
depression cs of the mercury column in b", and is
seen to be greater than that of water vapor. If
now a few drops of ether are introduced into the
tube 6', the depression of the mercury there is seen
to be equal to ct -{■ cs: i.e., the pressure of the ether
vapor is the same as before, and so also is that
of the water vapor. The total pressure, due to
both, is simply the sum of the pressures which
each would exert separately. So we see that the
pressure of each is the same as it would be if the
other were not present in the space.

Now incline the tube b". The mercury compresses the ether
vapor in it. The vapor now occupies a smaller volume, so some
of it must have been condensed into liquid alcohol, but the mer- •
cury still remains at the level s as before. The pressure cs of the
ether vapor is therefore unchanged, although its volume has
been much diminished. If the tube be held erect and lifted a
little way (but not out of the mercury in the vessel), the mer-
cury is seen to remain at the same level s. This time the volume
has been increased, but the pressure remains constant, showing
that some more ether must have evaporated to fill the space
and exert its pressure of saturation.

By warming or cooling the .vapor-filled space in either 6' or 6",
it may easily be observed that the pressure of saturation is greater
when the temperature is higher, and less when the temperature is
lower. The relations between temperature and pressure of

Fio. 84

148

PHYSICS

saturated vapor, as determined for water and for alcohol, are shown
in the curves (Fig. 85). The abscissas represent temperatures,
and the ordinates pressures in cm of mercury. The numbers on
the vertical scale, when multiplied by 10, represent cm of pressure;
those on the horizontal scale, when multiplied by 10, represent
degrees C. What pressure does the line Ap represent?

129. Boiling Point. From the curve W, tell what pressure is
exerted by saturated water vapor at a temperature of 20°, of 50°,
of 80°, of 100°. Do you note any relation between the pressure

corresponding to 100° and the
normal barometer pressure, 76
cm? What pressure does alco-
hol vapor exert at 20° (curve A),
at 50°, at 78°? Do you note
any relation between the pres-
sure corresponding to 78° and
the normal barometer pressure?
Since at 76 cm pressure, water
boils at 100° C. and alcohol at
78° C, may we then define
boiling point as the temperature
at which the pressure of satur-
ated vapor is equal to the sur-
rounding pressure? Suppose the
atmospheric pressure to be 42
cm, as on the* top of Mt. Blanc, at what temperature would water
boil there? At what temperature would alcohol boil there?

Those who have* answered correctly the questions just asked
will understand that the boiling point of any liquid may be defined
as the temperature at which the pressure of its saturated vapor is equal
to the surrounding pressure. Thus we see that a liquid may boil
at almost any temperature, since a reduction of the external
pressure lowers the boiling point and an increase in that pressure
raises it. For water, this change in boiling point corresponding
to a change of 1 cm in the barometric pressure is 0.37° C. For
example, when the barometric pressure falls from 76 cm to 74 cm

Fig. 85. Relation op PHessure of
Saturated Vapor to Temperature
for Water and Alcohol.

HEAT 149

a correct thermometer, placed in the steam of boiling water, will
read 100° - 2 X .37 - 99°. 26 C.

That the boiling point is the temperature at which the pressure
of the saturated vapor is equal to the surrounding pressure may
be readily appreciated from the common sense point of view. For
it is plain that ebullition (i.e., boiling) is different from evaporation
in that the steam escapes in bubbles from the midst of the liquid
instead of from the surface only. Now if the surrounding pressure
were greater than the pressure of the steam in these bubbles, the
bubbles would be unable to expand and float to the surface. On
the other hand, if the external pressure were less chan that of the
steam composing the bubbles, the water would flash into steam
instantaneously, as it sometimes does with explosive violence
when a defective boiler gives way.

130. Superheated Vapor. Let us now suppose that we have
a very small quantity of water in a closed fruit jar at a given
temperature. As has just been stated, the vapor will soon become
saturated and will exert the pressure that corresponds to this
temperature. If, now, we increase the temperature, more of the
water will evaporate, and the pressure of the saturated vapor will
increase to that corresponding to this higher temperature. Let
us now suppose that at this higher temperature all of the water has
been evaporated; what will be the effect of a further increase in
temperature? The pressure will increase, of course, but will it
increase as fast as it would if more liquid were present so that more
vapor would be formed? Clearly not. Therefore, when a satu-
rated vapor not in contact with its liquid is heated in a closed vessel,
its pressure at the higher temperature is less than that which it
would exert if it remained in contact with its liquid so that it con-
tinued to be saturated. A vapor not in contact with its liquid and
at a temperature higher than that corresponding to saturation, is
said to be superheated.

131. A Gas is a Superheated Vapor. Let us now consider
how the pressure of a superheated vapor varies with the tempera-
ture. So long as the vapor is superheated, none of it will condense,

150 PHYSICS

and there will be no liquid in the space, so that no more vapor can
be formed therein and none of the liquid can exist. Therefore
the changes in pressure produced by changes of temperature will
not be complicated by evaporation and condensation. Hence we
may surmise that such a vapor will behave very much like a gas.
Experiment has shown that superheated vapors, when they are
not too near the saturation point, do act in accordance with the
gas laws of Gay-Lussac and of Boyle. We therefore conclude
that a superheated vapor is really a gas. The converse of this con-
clusion, i.e., that a gas is a superheated vapor, has been verified by
the reduction to liquids of the so-called permanent gases, hydro-
gen, oxygen, and nitrogen.

132. Critical Temperature. We have seen that water can
exist as a liquid at all ordinary temperatures. Therefore at all
such temperatures superheated water vapor can be condensed
to a saturated vapor or to a liquid by the application of pressure
alone. On the other hand, the so-called permanent gases do not
exist at ordinary temperatures as liquids, and we find that we can
not, by any amount of pressure, condense them to liquids without
also cooling them. The temperature to which a gas must be
cooled before it can be converted into a liquid, is different for dif-
ferent gases, and is called the critical temperature. The crit-
ical temperature of water is 365^ C; that of air, — 140° C. Other,
critical temperatures are: alcohol, 243° C; ether, 194° C; ammo-
nia, 130° C; carbon dioxide, 31° C; oxygen, — 119° C; hydrogen,
-242°C.

From what has just been said we learn that we can not condense
a gas or a superheated vapor into a liquid by applying pressure
only, if the temperature is above the critical value for that gas. This
important fact is of far-reaching moment in the economy of nature,
as a study of the preceding figures will show. We note that for
water this critical temperature is high, so that at all ordinary tem-
peratures water exists as a liquid : the same is true of alcohol. On
the other hand, the critical temperature of air is very low. Hence,
at all ordinary temperatures air is a superheated vapor or gas,
and can not be liquefied by pressure alone: the same is true of hy-

HEAT

151

drogen and oxygen. Hence, we see why such low temperatures are
required for liquefaction of these gases. We can readily under-
stand how fortunate it is for beings organized as we are that these
substances are so en-
dowed; for if the crit-
ical temperature of
water were low, while
that of air were high,
we would know water
at ordinary tempera-
tures only as a gas, and
air under like condi-
tions largely as a liquid.
The entire economy of
nature would thus be
overturned; for what could we do with liquid air to breathe and
gaseous water to drink?

Fig. 86. Water Changes to Invisible Vapor
Which Condenses into Clouds

133. Humidity. The relations we have just been studying
act favorably to the maintenance of life on the earth in other impor-
tant ways. Thus, the fact that at ordinary temperatures water and
its vapor exist together shows us how it is possible for the water of
the ocean to evaporate and be carried in the form of vapor over the
land, to be deposited there as rain. We see also why there is always
considerable water vapor in the air. This humidity of the air

is an important factor
in climate. Every one
knows how oppressive a
hot day is if the hu-
midity is high; i.e., if
the water vapor in the
air exerts a pressure
nearly equal to that of
saturation at the tem-
perature of the air. Under these circumstances it is plain that
very little water can evaporate; and therefore we are not cooled
by evaporation from our bodies.

Fig. 87. Rain Clouds Deposit the
Water on the Land

152 PHYSICS

134. The Formation of Dew. The formation of dew is a
familiar phenomenon. Drops of water appear on the outside of a
pitcher of ice water on a warm day, because the temperature of the
pitcher is below that at which the water vapor in the air would
be saturated; i.e., below the dew point. Thus, by dew point is
meant the terrvperature to which the air mibst he cooled in order to
bring the water vapor in it to saturation, so that condensation begins.
Since the amount of water vapor in the air is of such great im-
portance to climate, its detennination is an important part of the
work of the Weather Bureau. What is termed relative humidity
is the ratio of the actual pressure of the water vapor in the air to
the pressure of saturated vapor at the same temperature.

. 135. Latent Heat. Another important fact about the con-
version of water into steam or into ice remains to be considered.
If we place a thermometer in a vessel of water and heat it gradually,
the thermometer indicates a gradually increasing temperature;
but when the water reaches the boiling point, although we continue
to heat it, the temperature remains constant until the change of state
is completed. Thus, when water is boiling under any given pressure
we can not raise the temperature of the water beyond the boiling
point that corresponds to that pressure. But what becomes of the
heat that is added after the boiling point is reached? It is used
in converting the water into steam; so that energy is required
to do this work. Is the quantity of heat thus required large?
Experiment shows that it is large, for it is found that 536 gm cal
are required to convert 1 gm of water at 100° into 1 gm of steam at
the same temperature. Since this amount of heat seems to disap-
pear in the process, it is called the latent heat of steam. Also,
conversely, when steam condenses into water, it gives up its latent
heat, every gram of steam returning its entire 53G gm cal.

A similar phenomenon accompanies melting and freezing,
but the amount of heat required' is not so great; thus it requires
80 gm cal of heat to convert one gm of ice at 0° C. to one gm of
water at the same temperature. Conversely, when water is frozen
it gives up this same quantity of heat per gm. Every substance
absorbs a definite amount of heat per gm while melting or evapo-

HEAT 153

rating and gives up this energy while solidifying or condensing, the
amount thus transformed being different for different substances.

136. Latent Heat is a Form of Energy. From the fore-
going discussion it must be quite clear that latent heat is a form
of energy, for heat energy is expended in doing the work of convert-
ing the liquid into the vapoi: form, and is given up again as heat
when the vapor is condensed into the liquid form.

We can now * understand why it is that when a substance is
vaporizing or condensing, or when it is liquefying or solidifying
under a constant pressure, its temperature remains constant
until the transformation is completed. For when heat energy.
is doing the work of changing the state or internal condition of the
substance, it can not at the same time be employed in raising the
temperature. Conversely, when a mass of vapor is liquefying,
each gm of vapor that condenses gives up its latent heat, so that
the temperature of the liquid can not fall so long as any vapor
remains to be condensed and supply it with heat. The case is
the same when liquids solidify.

137. Water and Climate. We can understand also why
water is so important in regulating atmospheric temperatures,
because its specific heat, and its latent heat of vaporization and of
solidification are so great. When water is warmed, or changed
from ice to water, or from water to vapor, it absorbs large quan-
tities of heat, and so prevents the atmosphere's heating as rapidly
as otherwise it would. Conversely, when it is cooled, or changed
from water to ice or from vapor to water, it gives out large quan-
tities of heat, so that the temperature of the atmosphere does not
fall as rapidly as otherwise it would. Since water is evaporated
in great quantities from the oceans and since some of it is then
carried with the winds over the land to be there condensed, it serves
the earth very much as a steam heating system serves our offices
and dwellings.

SUMMARY

1. Zero temperature is that of melting ice.

2. Unit temperature interval is the Centigrade degree. This
is the j^ part of the interval between the temperatures of melting
ice and water boiling at 76 cm barometer pressure.

154 PHYSICS

3. When the pressure remains constant, a given mass of gas ex-
pands 2I3 of its volume at 0° C, for every increase in tempera-
ture of 1° C. (Gay-Lussac's Law.)

4. Absolute temperature is equal to 273° + Centigrade tem-
perature.

5. When its volume remains constant, the pressure exerted
by a gas is proportional to its absolute temperature.

6. Unit quantity of heat is the gram calorie, i.e., the quantity
of heat involved in changing the temperature of 1 gm of water
1° C. Its symbol is gm cal.

7. Specific heat of a substance equals the number of calories
absorbed by. one gram of it in warming 1° C.

8. The total number of calories absorbed or given off by any
body during any change of temperature = specific heat X mass X
change of temperature.

9. Every saturated vapor exerts a pressure that depends on its
temperature only.

10. A gas is a superheated vapor.

11. Gases can not be condensed into liquids by pressure, however
great, at temperatures above the critical temperature.

12. Water vapor is an important constituent of the earth's
atmosphere.

13. The dew point is the temperature at which the water vapor
in the atmosphere becomes saturated.

14. The relative humidity of the atmosphere is the ratio of the
pressure of saturated vapor at the dew point, to its pressure at the
temperature of the atmosphere.

15. Heat is absorbed during the processes of melting and
evaporation, and given out during the converse processes of solidifi-
cation and condensation. (Latent heat.)

16. 80 gm cal of heat become latent heat when one gm of ice
melts; and 536 gm cal, when one gm of water evaporates at
100° C.

17. When a body changes state, the number of calories absorbed
or given off by it is equal to the corresponding latent heat, multi-
plied by the number of grams mass.

18. Latent heat is a form of energy.

HEAT 155

QUESTIONS

1. How far can we rely on our sense of touch for information con-
cerning temperature? Give some examples.

2. What elements are necessary in determining a temperature
scale, and how is the Centigrade temperature scale defined?

3. Why is Galileo's air thermometer inaccurate? What device is
employed to keep either the volume or the pressure of the air con-
stant?

4. What other instrument must be used in connection with an air
thermometer in determining temperature? Why?

5. How much does a gas expand when heated 1°C.? What is Gay-
Lussac's law?

6. What do we mean by absolute temperature?

7. Is there any relation between the pressure of a gas at constant
volume and its absolute temperature? What is the relation?

8. How do we define the unit quantity of heat?

9. How do we compare heat quantities? What is specific
heat?

10. Does water vaporize at all temperatures?

11. If we have water vapor in contact with its liquid in a closed
vessel, is there any limit to the pressure it can exert at a given tem-
perature?

12. When is a vapor said to be saturated?

13. Does the pressure that a saturated vapor in a closed vessel
exerts depend on the volume of the vapor, on the pressure of the air
or other substances in the vessel, or on anything but the tempera-
ture?

14. Is a vapor in contact with its liquid in a closed vessel always
saturated?

15. When is a vapor superheated?

16. Compare the properties of a superheated vapor with those of a
gas.

17. When can we condense a superheated vapor to saturation by
compression alone? When is cooling also necessary?

18. What do we mean by critical temperature? Is it the same for
all substances? Why is it fortunate for animal and vegetable life
that the critical temperature of water is high while that of air is low?

19. In what way does the large heat-absorbing power of water act
favorably on the climate of places near large bodies of water?

20. Under what conditions does dew settle on an object?

•21. What is meant by relative humidity of the atmosphere?
22. If only a small quantity of heat were to become latent heat
when water passes into water vapor, what sort of climate would the
earth have?

156 PHYSICS

PROBLEMS

1. The freezing temperature of water is marked 32° on the Fahren-
heit scale, the boiling temperature 212°, and each degree on this scale
corresponds to a temperature interval of | of a degree on the Centigrade
scale, (a) The normal temperature of the human body is 98.4° Fah-
renheit. How many Fahrenheit degrees is this above the freezing point
of water? To what reading on the Centigrade scale does it corre-
spond? To what on the absolute scale? (6) When the temperature
of a school-room is 70° Fahrenheit, what will a Centigrade thermom-
eter read? (c) Show that the following rule is correct: To find the
Centigrade reading that corresponds to any Fahrenheit reading, sub-
tract 32 from it, and multiply by J. Make up a rule for changing Cen-
tigrade readings to Fahrenheit, (d) Mercury freezes at —38.8° C.
What is the freezing point of mercury on the Fahrenheit scale?

2. A student in the chemical laboratory collects 156 cm^ of oxygen
gas at 20° C. and 78 cm barometric pressure. What would its volume
be at 0° C. and 76 cm pressure? Its density under the latter condi-
tions is 0.00143 151 What is its mass?

cm*

3. How many cm' of air are there in a schoolroom whose dimensions
are lO X 15 X 5 m? The density of air at 0° C. and 76 cm pressure
being 0.00129 ^^, what is its mass? Its specific heat being 0.237,' how
many calories of heat are required to raise its temperature from (.° to
20° C?

4. Suppose the air in the room, problem 3,. must be completely
changed every 15 minutes, how many calories are required each hour?
If this heat is to be taken from hot water, which enters the radiators
at 85° and leaves them at 76°, how many gm of water must be delivered
from the boiler each hour for this room? One gm of coal when
burned gives up 7500 gm cal. How many Kg of coal are required per
hour if 50 per cent of the heat of the coal gets to the radiators?

5. When 100 gm of lead shot at 99° are mixed with 25 gm of
water at 5°, to what temperature /will both come, if the specific heat
of lead is 0.033?

6. The specific heat of steam is 0.48, of ice, 0.505. Steam at 105°
is mixed with 10 gm ice at — 10°, and the temperature is found to be 40°.
How many calories were required to warm the ice to 0°? To melt it?
To warm the resulting water to 40°? How much heat did each gm of
the steam give up in cooling to 100°? In condensing to water? How
many calories did each gram of this condensed steam give up in cooling
to 40°? Let m represent the whole mass of the steam, and write 'the
expression for the whole quantity of heat given up by the steam in
changing from steam at 105° to water at 40°. Supposing no heat
entered or left the vessel in which they were mixed, how must this

HEAT 157

aipaount have compared with that absorbed by the ice in warming
to 0°, melting to water, and coming to 40°? Find the value of m, the
mass of steam used.

7. If 100 gm water at 50° are mixed with 200 gm ice at 0°, will all
the ice be melted? If not, how much? If so, what will be the result-
ing temperature? Answer the same questions if the mass of the water
was 500 gm, and its temperature 80°.

8. An iron girder bridge is 30 m long when its temperature is —10^
C. Taking the mean coefficient of expansion of iron as 0.000012, what
is the length of the bridge when its temperature is 37° C?

9. A bottle contains 2500 cmSof alcohol at 20° C; what will be
the volume of this alcohol at 0° C? The coefficient of cubical expan-
sion of alcohol is 0.0011.

SUGGESTIONS TO STUDENTS

1. Repeat the experiment described in Art. 117. Have you no-
ticed a similar phenomenon when going from a cold room to one of
medium temperature and comparing your sensations with those of
your schoolmates who have come from a room that is overheated?

2. How is water purified by distillation? Petroleum consists of a
number of different . components, each having its own boiling point;
suggest a method for separating these components.

3. Examine the pendulum of a regulator clock at a watchmaker's,
and see if you can find out how the downward expansion of the pen-
dulum rod is compensated by the upward expansion of another metal.

4. Lay a thin strip of brass on a similar strip of iron, rivet the two
. together, throw the combination into a fire, and see what it will do.

Examine the balance wheel of your watch; does your observation on
the brass-iron strip help you to explain how the watch is compensated?

5. Examine the device (thermostat) by which the temperature of
an incubator is kept constant. Make a diagram of it and report.

6. How does a wheelwright put an iron tire on a wheel so that it
will be tight?

CHAPTER VIII
TRANSFER OF HEAT

138. Conduction and Convection. In the preceding chapter,
wheii heating and cooling were mentioned it was assumed that
these terms would be understood. It may be well before wo leave
the subject to give fixed form to our ideas concerning these
processes. When we are heating water in a tea-kettle, we notice
that before it can reach the water, the heat must first pass through
the copper bottom of the kettle. Further, those portions of the
water that are nearest the fire must become heated first. How
do they transfer this heat to other portions of the water? Why
does covering boiler or a steam pipe with an asbestos coating pre-
vent waste of heat?

In considering these questions, the first point to note is that
when we wish to impart heat to a substance we bring it near or in
contact with something hotter; and conversely, when we wish to cool
it we place it near something colder. This almost instinctive prac-
tice is based on the universally accepted concept that heat in some
way passes from the hotter body to the cooler, and not the reverse.
But is this strictly true? Do we never find cases in which heat
passes from a cooler to a hotter body? Before answering these
questions we must distinguish among several different kinds of
heat transference.

One form of heat transference is illustrated in the passage of
heat through the bottom of a kettle or along a solid rod when
one end is heated in a flame. In this case the particles of
the kettle or the rod do not change their relative positions,
but merely pass the heat along from particle to particle. This
form of heat transfer is called conduction, and it is the process
by which heat moves from one part of a solid to another. In con-
duction, heat always flows from portions at higher temperatures
to others at lower temperatures.

168

TRANSFER OF HEAT

159

In the case of liquids and gases, however, the process of heat
transfer is somewhat different. All fluids are very poor conductors
of heat, but when a small portion of the fluid becomes warmed
by conduction from a heated body, it
expands, and hence becomes less dense
than the surrounding portions of the fluid.
It is therefore pushed upward by those
heavier surrounding portions, which
creep in below; and as it goes it carries
its heat with it. Thus currents are set
up in the fluid, the cooler portions set-
tling downward and pushing the warmer
portions upward.

The name of this process is con-
vection. It is the process by which
heat usually spreads through liquids and
gases, and it continues as long as there
is any difference in temperature between
the different parts of the fluid. This
process is illustrated in Fig. 88, as it
takes place in a chimney. The smoke I

shows how the hot gases flow away at Fio. 88. The Heavy Fluid
. , . J iu i_ i_ i.i_ Displaces the Lighter

the top, and the arrows show how the

cold air pushes its way in below, forming a so-called "draft" along

the floor and up the chimney.

139. Applications of Condnction and Convection. The

knowledge of the conducting powers of different substances is
very useful in daily life. The asbestos coverings on locomotive
boilers and steam pipes are poor conductors of heat, and so pre-
vent its escape. So does wool, whether in its natural state on a
sheep's back, or as cloth in our garments. Air conducts very little
heat ; hence the air spaces between the walls of refrigerators. Down
comforters are very warm for their weight, because the air, entangled
with the down, keeps the heat from escaping. Water is a very
bad conductor of heat; hence it must be heated from below, in order
that convection currents may start in it. The circulation of air

160 PHYSICS

in a hot-air heating system, and of hot water in a hot-water system,
is in many cases secured entirely by convection. The drafts up
a lamp or factory chimney are not essentially different from trop-
ical whirlwinds.

140. Radiation. In both of the processes just described the
substance heated must be in contact with the hotter body. Now
there must be some other process of heat transfer; for we all know
that a hot stove will heat objects in its vicinity, though not in con-
tact with them, to a temperature higher than that of the surround-
ing air, and that the life-giving energy of the sun somehow suc-
ceeds in reaching the earth without any apparent contact between
the two. Hence, for describing this process of heat transfer, we
are compelled to imagine another mechanism which is called

RADIATION.

141. Diffusion. We can most easily gain some conception
as to how radiation takes place, if we pause for a moment
to consider some other phenomena which will enable us to form
an idea as to what heat is. When any gas having an easily recog-
nizable odor, such as illuminating gas, is liberated in a room,
in a very short time the odor can be detected in every part of the
room. This familiar fact justifies us in concluding that the gas
has spread throughout the entire space. How may this have hap-
pened? Evidently the gas must consist of numerous particles, and
these particles must have moved from the place where they were
liberated into all other portions of the room. Hence we are led
to think that the particles of this substance must have been in
motion before they were liberated, and that to liberate them it was
merely necessary to open a stopcock, or take out a cork, so as to
provide an opening in the confining walls, through which the gas
particles might escape. The simplest idea that we can form of
their motion is that each particle is highly elastic and that it con-
tinues to move in a straight line until it collides with some other
particle or with the sides of the vessel; when it immediately re-
bounds and starts off again in a new straight path.

TRANSFER OF HEAT ,161

142. Evaporation. Let us now consider whether this
hypothesis will help us to a better understanding of the phenomena
of evaporation, of which we learned in the preceding chapter.
We there found that water evaporates at all temperatures, i;e., the
water particles fly away from the liquid surface and diffuse them-
selves into the surrounding air. If we assume that the water
particles are in violent motion while within the body of the liquid,
we can form a mental picture of how the particles that are at the
surface might be more free to fly out into the air than to fly back
into thQ liquid. Further, since a vapor always occupies so much
more space than does the liquid from which it has been formed,
we must conclude that the particles of water vapor are farther
apart than are those of liquid water, and therefore we can under-
stand why they have far greater freedom of motion.

143. Diffusion of Solids. Again, we are familiar with what
often happens when we put a solid into a liquid. For example, a
lump of sugar, placed in a cup of coffee, disappears after a time, even
without stirring; and, if left long enough, it will sweeten all
the coffee in the cup. Here again we are led to conclude that the
sugar and the liquid consist each of a large number of particles which
are already in motion, and that when the solid is put into the liquid,
the moving particles of each spread themselves amongst those
of the other. When the particles of two or more substances thus
spontaneously mix together, they are said to diffuse into one another,
and the process is called diffusion.

144. Gaseous Pressure. With the aid of our hypothesis
we can now get a very satisfactory conception of the manner in
which a body of gas exerts a jpressure on the walls of a vessel in
Vhich it is confined. For it is easy to see that if the millions of tiny
gaseous particles are flying with great velocities in all directions,
the sides of the vessel will be struck at every instant by a multitude
of these particles, and that each of these when it strikes and
rebounds will give the wall a push or impulse. The magnitude
of this impulse will depend on the mass and velocity of the particle
(c/. Art. 39). The sides of the vessel will thus be bombarded

162 . PHYSICS

at every instant by a multitude of the little particles, which come
so thick and so fast that the sum of their impulses can not be dis-
tinguished from a continuous pressure.

145. Effect of Heating. We have learned, (Art. 121) that
when a gas is heated in a closed space at constant volume, the
pressure that it exerts is increased; but we have just seen that the
pressure depends both on the masses and on the velocities of the
gaseous particles. Hence, since heating the gas does not change
the mass of the flying particles, it must increase their velocities.
Also, since at a given instant each particle has a certain mass ?m,

and a certain velocity v, it has a certain amount of kinetic energy — ;

and since, as we have just found, heating increases the velocity
t;, we are forced to conclude that what the heat energy does when
it raises the temperature is simply to increase the kinetic energy
of the little particles. Thus we are led to infer that heal is nothing
more nor less than the kinetic energy of these moving particles,

146. The Kinetic Hjrpothesis. The hypothesis at which
we have now arrived includes the following ideas:

1. Every substance consists of a great number of very small par-
ticles, each of a definite mass. These particles are called mole-
cules.

2. These molecules are constantly in rapid motion.

3. Heat is the kinetic energy of these moving molecules,

4. The temperature of a body depends on the average kinetic
energy of the individual molecules of the mass, while the total quxm-
tity of heat possessed by it depends on the sum of the kinetic energies
of all its molecules.

This kinetic hypothesis has helped us to a better understanding
of diffusion, evaporation, and gaseous pressure; let us now return
to radiation and see if it will assist us there.

147. Radiation. We have learned that radiation consists in
the transfer of energy from one body to another when the two are
not in contact, as, for instance, from a stove to your hand. Now,

TRANSFER OF HEAT 163

if the particles of the stove are in rapid motion, and if heating
the hand consists in making its particles move more rapidly,
by what possible mechanism may t6e rapidly moving particles
of hot iron communicate some of their motion to the particles
of your hand across the intervening space? Does anything like
this happen when a stone is thrown into a pool? Does not the

Fig. 89. Energy May be Transmitted by Waves

motion of the stone produce waves, as sho^vn in the photograph
(Fig. 89), which move in gradually widening circles until they
reach the borders of the pool? What happens to the pebbles there
when the waves reach them, and what becomes of the waves
themselves? Do they not set the pebbles in motion, thus passing
along to the pebbles the energy that they received from the stone?
May we not, then, imagine that just as water waves spread out in
rings from the spot where a stone falls, so heating waves spread out
in all directions from the vibrating molecules of hot bodies; and
just as water waves break and give up their energy to pebbles on
the shore of the pond, so the heating waves strike and give up
their energy to particles on the boundaries of their realm?

148. The Ether. In the case of the pebbles that are set
in motion when a stone is thrown into a pool, it is evident that
water is the medium through which the energy travels, and that
without some such medium, no energy can be thus transferred.
Now if heat is transferred by waves, what is the medium in which

164 PHYSICS

these waves are propagated? That it is not air, nor ordinary
matter of any kind, must be manifest to any one who will but hold
his hand near an electric glow lamp; for the air has been pumped
out of the bulb, yet the filament radiates both heat and light through
this vacuum. So also we are warmed by the energy that comes
to us from the sun, although there are good reasons for believing
that the greater part of the space between does not contain any sen-
sible amount of ordinary matter. Hence the adoption of the wave
hypothesis for radiation makes it necessary to assume that there
exists in space a medium which is not ordinary matter, but which is
capable of transmitting such waves. This medium is called the
ETHER. When we come to study electricity and light, we shall
meet with it again, and, are likely to become more deeply im-
pressed with the utility of the ether-wave hypothesis.

149. Prevost's Theory of Exchanges. Suppose that the
fire goes out so that the temperature of the stove falls to that
of the hand; then you can no longer warm it at the stove. Has
the stove then ceased to send out heat waves? If now we bring
a piece of ice near the stove, will not some of it melt? Yet when
the stove has further cooled to the temperature of the ice, this latter
will no longer be melted by the heat from the stove. Has the stove
then ceased to send out heat waves? Is it not simpler to suppose
that all bodies are radiating heat waves at all temperatures, and
that whether a body grows hotter or colder depends on whether,
in a given time, it absorbs more than it radiates, or radiates more
than it absorbs?

We may state this theory as follows: All bodies are radiating
and absorbing heat energy at all times. If, in a given time, the
amount of energy that a body absorbs is greater than that which
it radiates, the temperature of tJie body rises; while if the amount
of energy that it absorbs is less than that which it radiates, its tem-
perature falls. This theory was first propounded by Prevost, and
hence it is known as Prevost's Theory of Exchanges.

160. Absorption. Radiant heat, though not absorbed by
the ether, is always absorbed to a greater or less degree by ordi-

TRANSFER OF HEAT W5

nary matter. Elaborate experiments have been made to determine
how much of the energy of radiant heat is absorbed by various
substances; and it has been found that of all the gaseous
substances, water vapor has the largest absorbing power. Thus
the experiments of John Tyndall showed that a layer of air satura-
ted with water vapor at ordinary temperatures, and four feet thick,
absorbs 20% of the radiant heat energy that falls upon it. Of
course solids and liquids absorb more of the radiant heat than this.

151. Absorbing Power of Water Vapor. The fact that water
vapor is a powerful absorber of radiant heat furnishes us with
another admirable example of nature's adaptation of means to
an end; for what would be the condition of the earth's surface

Fig. 90. Glacier and Snow-Field on a High Mountain

if the water vapor did not absorb a large portion of the sun's
energy? We should be exposed to a blistering heat in the day-
time, and a freezing temperature at night. That this is actu-
ally the case on high mountains, where the earth is not so thickly
blanketed with water vapor, is well known to every one. This
is largely due to the fact that water vapor is transparent to
light, i.e., it absorbs very little of the sun's energy that reaches
it in that form; while on the other hand it absorbs a very large
proportion of the radiant heat waves. When the light energy
from the sun has passed through the vapor-laden atmosphere, it
is absorbed by the bodies on the surface of the earth, and is there
converted into heat. At night, when these bodies arfe sending out
this energy as radiant heat, and are radiating more energy than
they are receiving, the water vapor in the atmosphere absorbs

166

PHYSICS

this radiant heat, and prevents it from escaping into space. Thus,

acting like a trap to catch and hold the sunbeams, the water vapor

accumulates heat energy in the day-time and

throughout the summer, and holds it over for the

nights and the winter.

The glass of a hothouse acts in a similar
way: it lets the light in; but when the light has
been absorbed and converted into heat, the glass
will not let it escape by radiation. So it remains
and keeps the plants warm. For this reason
vegetables can be grown under glass very early
in the spring-time without the aid of artificial heat.

Fig. 91. Automo-
bile Cooler

152. Radiation and Absorption. A close relation is found to
exist between radiation and absorption. Substances that send out
large amounts of radiant heat and light when red hot, are found
also to absorb large quantities when cold. Thus, a substance like
lampblack is dark when you look at it because it absorbs nearly
all the light that falls on it;
it is also found to be a pow-
erful absorber of radiant
heat. We all know well that
this substance is a splendid
radiator, for carbon is used
in the manufacture of elec-
tric lights, and is present in
large quantities in oil and
gas flames. Furthermore,
coal and wood-charcoal,
which are also forms of car-
bon, when burning become
powerful radiators of both
heat and light.

Since radiation and ab-
sorption take place only at the surfaces of bodies, it follows other
things being equal, that the greater the surface the greater is the
radiation or absorption. Hence radiators for heating houses and

Fig. 92. Detail Fig. 91 Showing Air Cells

AND WaTER-PiPES WITH LaRGE RADIAT-
ING Surface

TRANSFER OF HEAT

167

for cooling automobile engines are better if black and rough than
if bright and polished, because then they radiate more outward and
reflect less inward for a given surface, and because when they are
rough they have a larger surface.

163. Heat and Light. When any substance is heated, as in a
blacksmith's forge, it becomes red-hot, at a temperature of 520° C,
i.e., it begins to give out red
light; and if it is further
heated, it not only gives out
more heat, but the light that
it emits becomes first yellow
and then white. Since
bodies at high temperatures
give out both radiant heat
and light, and since both ra-
diant heat and light are con-
verted into heat when they
are absorbed, it appears that
they must be closely related
phenomena. It is, therefore,
reasonable to suppose that if
the radiant heat is a wave
motion, light is also a wave
motion. If this conjecture is
correct, how does light differ
from radiant heat?

We may imagine that, just as on a calm day we have the
long, steady roll of the ocean, so from bodies at low temperatures
we have long heat waves; and on the other hand, just as, during
a storm, we have not only that same long roll of the ocean much
intensified, but also a multitude of shorter waves added to it, so
from red-hot bodies we have not only the long radiant heat waves
intensified, but also shorter waves which give us the sensation of
light. We shall have occasion to test tliis assumption in a num-
ber of ways when we come to take up the detailed study of light
in the last five chapters of this book.

Fig. 93. The Hot Iron Sends Out Waves
OF Various Lengths

168 PHYSICS

SUMMARY

1. Heat transference is of three kinds, conduction, convection,
and radiation.

2. The facts of diffusion, evaporation, and gaseous pressure
lead to the hypothesis that heat is molecular kinetic energy.

3. The facts of radiation suggest the hypothesis that radiant

Fig. 94. The Big Roller Carries Smaller Waves on its Back

heat and light are forms of wave motion in a medium called the
ether.

4. All substances are radiating heat waves at all temperatures.

5. A body does not change its temperature when it receives
in a given time as much radiant energy as it itself sends out.

6. The amount of radiation depends: 1, on the radiating
substance; 2, on the difference in temperature between the radiating
body and the surrounding space; 3, on the amount of surface
exposed.

QUESTIONS

1. How do we describe the process by which heat travels along
an iron rod when one end of it is held in a flame? What name do we
give to the process?

2. What can you say of the conductivities of different substances,
solid, liquid, and gaseous?

3. What is the name of the process of heat transfer in fluids? De-
scribe the process, and show how it differs from conduction.

4. When bodies are not in contact, how do we conceive that heaJb
travels from one to the other? What is the name of this process?

5. When two bodies not in contact have the same temperature,
what must be their action with regard to radiation?

6. Why do we suppose that all bodies are sending out radiant
heat waves at all temperatures?

TRANSFER OF HEAT 169

7. Do all bodies at the same temperature have the same radiating
power? What ones radiate most?

8. What relation has the radiating power of a substance to its
absorbing power?

9. What are some of the conditions that are favorable to rapid
radiation?

10. Mention some practical applications of conduction, convection,
and radiation.

PROBLEMS

1. Why are wooden handles put on short fire pokers, but not on
long ones?

2. At what level should the cool water return pipe and the cold
water supply pipe enter the hot water reservoir of a water heating
system operated by convection? From what level should the hot
water main leave? Should the heating pipe in the stove or furnace
be horizontal or inclined? Which should be at the higher end, the
cold water supply pipe or the hot water delivery pipe? Give reasons.

3. How are both evaporation and diffusion illustrated by liquid
perfumes? Do solid perfumes act in the same way?

4. Do clothes dry while frozen, when they are himg out on the line
in freezing weather? Does snow ever disappear from the ground
without melting? Do crystals ever form on the sides of a bottle of
gum camphor? Do solids never pass from the solid to the vapor state
without first passing into liquid?

5. Do you become warmer in summer if you wear a black suit than
if you wear a white one of the same material? Why? Why is loosely
woven, thin material more comfortable to wear in summer than thick,
compact material?

6. What is the use of tarred paper in the walls of houses? In what
way do double windows save heat?

7. By what process or processes is heat distributed, when a room
is heated by hot water radiators? By steam radiators? By indirect
radiators? By hot air registers? By stoves? By grates? Can you
get hot air to come out of a register into a room that is tightly closed?

8. What advantage arises in the hot water heating system from the
high specific hect of water? In the steam system, from the high latent
heat of steam?

9. Why does a room become so very hot in summer when the sun
shines into it through a closed window?

10. Sketch an arrangement for cooling a house by means of a fur-
nace which is to consume ice instead of coal, and send cold air instead
of hot air out of the registers. Sketch a similar scheme for reversing

170 PHYSICS

the action of a hot water heating system and an ordinary stovcf, speci-
fying in each case the locf^tion of the cooler in the house.

11. Why is the word *' draft" an inappropriate one to use in connec-
tion with convection?

SUGGESTIONS TO STUDENTS

1. With a thermometer, take the temperature of carpet, oil-cloth,
metals, and wood in a cold room. Feel each of the substances with
the hand. Explain why the sensations do not agree with the indica-
tions of the thermometer. Make the same experiment with several
substances after warming them in an oven.

2. Stir some scrapings of blotting paper into a glass of water, add
a lump of ice, and make a diagram of the convection currents that are
shown by the scrapings. Make the same experiment, using a hot clay
marble instead of the ice.

3. Examine a refrigerator. Do you find any provision for securing
a circulation of air in it by convection? If so give diagram and brief
description.

4. Investigate a hot air heating system, and hand in a diagram
with a brief explanation of how the air circulates, and how it heats the
rooms. Do the same for a hot water system and a steam system
Visit the stores where such heaters are on sale, and ask for information
illustrated circulars, etc. Investigate particularly the heating system
in your own home, and give in a brief paper the results of your ex-
perience as to the best methods of managing it, especially the regula-
tion of the cold air supply.

5. Read the chapter on Heat in Experimental Science^ by Geo.
M. Hopkins (Munn & Co., N. Y.). Many interesting and easy
experiments are described in it.

6. Tyndall's Heat as a Mode of Motion (Appleton, N. Y.)
contains the descriptions of many interesting experiments in conduc-
tion, convection, radiation, and absorption. Much of the theory of
this book is, not up-to-date, but the facts^ told in TyndalPs charming
and masterly style, are well worth reading. Some of the experiments
may easily be repeated.

CHAPTER IX
HEAT AND WORK

154. The Mechanical Equivalent of Heat. Let us now pass
finally to the consideration of the relations between heat and
mechanical work. Nearly everybody has polished a cent by rub-
bing it on a carpet, or seen sparks fly from a grindstone when a
tool is being ground, and therefore knows that heat can be pro-
duced by mechanical work. No definite relation between work
and heat was recognized in the early days of science. When Sir
Humphry Davy, in 1810, showed that he could melt two pieces
of ice simply by rubbing them together, the idea that heat could
be produced by mechanical work began to prevail.

Since heat can be converted into work, the question at once
arises, how much work can one gram calorie of heat do? The
first to solve this problem experimentally was James Prescott
Joule (1818-1889), who attained the result in a very interesting
and instructive way. His apparatus is pictured in Fig. 95, and
consisted of a calorimeter filled with water, in which paddles were
made to rotate by the weight of a falling mass. Thus the energy
of the falling mass is expended in heating the water by friction.
The amount of the work done is measured by the product of the
weight of the mass and the distance through which it falls. The
number of calories of heatv generated is the numerical product of
the specific heat of the water, its mass, and the change of tem-
perature {cf. Art. 126). More recently Rowland, at Johns Hop-
kins University, made a most careful and accurate determination
of this constant. The method employed by him was not different
in principle from Joule's. The result of his experiments has been
adopted as the jnost probable value of the number of ergs that
is equal to one gram calorie. It is

1 gm cal = 4.19 X 10^ ergs. (9)

This ratio is known as joule's equivalent, or the mechanical

EQUIVALENT OF HEAT.

171

172

PHYSICS

155. A Gas is Heated When it is Compressed. Who has not
noticed that a bicycle pump becomes heated when it is being used
to pump up tires? Since a great deal more heat is thus developed
than can be attributed to friction, we are forced to conclude that

the greater part of it is produced by
the work done in compressing the
air. This is perhaps the most
direct and simple case of the trans-
formation of energy into heat.

®With the help of the kinetic
hypothesis, which we adopted in
the preceding chapter, we may eas-
ily form a mental picture of the way
in which the heat is added to the
gas when it is compressed. For if
the piston of the pump is moving
downward, while millions of the
little molecules are flying upward
against it, each little molecule Will
rebound with an increased veloc-
ity, just as a base ball rebounds
from a moving bat. But this in-
creased velocity implies an increase
in the kinetic energy of the mole-
cules; and the total kinetic energy
that has been thus added to all the
molecules — i.e., the added heat — is the equivalent of the work
done by the advancing piston in compressing the gas.

Fig. 95. Joule's Apparatus

a. Revolving Blades.

b. Stationary Blades.

166. A Gas Cools When it Expands. That a gas cools when
it expands also follows at once from our hypothesis. For just as a
base ball strikes against the hands of the catcher and gives up its
kinetic energy in putting them into motion, so the little molecules,
when they bombard the piston so as to put it into motion, must
give up some of their kinetic energy in doing this work. The total
kinetic energy thus lost by the molecules — i.e., the lost heat — is
the equivalent of the work done by the gas on the piston.

HEAT AND WORK 173

It will make no difference whether the piston is present or
not; for if a compressed gas is allowed to expand directly into the
air, it must do work in pushing aside the air; and it must lose
heat in order to do this work. Doubtless many of you have made
an experiment that verifies this prediction; for you may have held
your hand near the valve of your bicycle tire when the compressed
air was escaping from it, and observed that the Jet of air seemed
very cold. You may also have noticed that when a bottle of
ginger ale or pop is opened, a cloud of condensed vapor appears
near its mouth. The gas escaping from the bottle becomes so
cold from doing the work of pushing away the air, that it cools
this air below the dew point. .

157. Liquid Air. This principle is the one that is used in
the liquefaction of gases, notably of air. The air is first cooled
and compressed as far as possible, and is then allowed to escape
through a small opening into the room. In this escape the at-
mosphere must be pushed back by the expanding air; and heat must,
therefore, be supplied to do this work. The only heat immediately
available is that still remaining in the compressed and cooled air.
So much heat is taken from it that part of it is condensed into a
liquid.

168. Cooling by Evaporation. In Chapter VII we learned
that a large 3,mount of heat must be supplied in order to change
a liquid into a vapor. We are now in a position to appreciate
that this heat is employed in doing the work of changing the
liquid into the vapor. We can also understand that if this heat
is not supplied from some external source, it may be obtained
from the liquid itself, in which case the temperature of the liquid
falls. When we recall the fact that the amount of heat required
to vaporize liquids is usually very large — ^in the case of water
for example, more than 500 gm calories per gm — we can see why
this principle may be advantageously employed in cooling proc-
esses. By the rapid evaporation of liquid air, temperatures
lower than —182° C. may be obtained.

174 PHYBICS

159. Manufactured Ice. Ice is now made in all large cities,
and in the process the principle just discussed is applied. Am-
monia or carbon dioxide gas is condensed into a liquid by means
of a powerful force pump driven by a steam engine or an electric
motor. The liquefied gas is then allowed to escape through a
valve into a system of pipes from which the air has been pumped.
These pipes pass back and forth in a large tank which is filled
\^ith salt water, so that the heat required for evaporating the
liquid carbon dioxide, is taken from this brine. This is then
cooled until its temperature is several degrees below 0° C; but
it does not freeze, because the freezing point of salt water is lower
than this. The pure water that is to be frozen is placed in large
iron molds, which are submerged in the cold salt water, and are
kept there until the water in them is frozen.

160. Cold Storage. Cold storage rooms are generally op-
erated in connection with artificial ice plants. These rooms have
thick walls made of nonconducting materials, and around them
on the inside are rows of pipes. The brine from the freezing
tank^ is pumped through these pipes on its way back to the cool-
ing tank, and thus serves to reduce the temperature of the room
to the point desired. Immense quantities of eggs, butter, fruit,
and other perishable foods are thus preserved in cold storage for
use in the winter months.

161. The Steam Engine. We are now prepared to consider
the way in which heat is converted into mechanical work by
steam engines. We select the locomotive as a typical case of
these 'engines, because it is complete in itself. Fig. 96 shows
the construction of a modern locomotive. The heat is derived
from a fire, which is built in the fire-box Fb at the rear end of the
boiler. In order that this heat may pass easily into the water in
the boiler, the smoke and hot gases produced by the fire are sent
through a large number of tubes Ft, which pass through the boiler
from the fire-box to the forward end. The water in the boiler
surrounds these tubes and the fire-box, so that it is in a position
to absorb a large part of the heat of the fire. When the engine

HEAT AND WORK

175

QQ H »^ •^

• ,■< r- p*

5? H ^ ►tj
<l> 3* f1" =•

'^ ?: c a-
2.^ cr Q

2. s; ^ a.
o 3 2 »

•a o

5 '^ w w

- ^ ? "

P c^ p c
o o ^ ^

•5'

gill

«> o

g§

176 PHYSICS

has "steam up/' the upper part of the boiler is filled with steam
at high pressure.

When the engineer wishes to start the engine, he pulls on a
lever called the throttle; and this lever opens the throttle valve

Tvy which is placed at the highest
point in the boiler. When this
valve is opened, the steam rushes
into the large supply pipe, which
conducts it to the steam chest Sc,
whence it passes to the cylinder C.
^'°- ^^* rSRw^ARr'' ^''^^''^ Having reached the cylinder through

the port p, the steam pushes against
the piston P and moves it backward. This motion of the piston
is transmitted by means of the piston rod Pr and the connecting
rod Cr to the driving wheels Dw, which are thus made to turn.
When the piston has reached the rear end of the cylinder, the
slide-valve automatically opens the port p', and also connects the
port p with the exhaust e, Fig. 97. The high pressure steam in the
steam chest then rushes into the rear end of the cylinder and
pushes the piston forward, driving the steam in the forward end
out through the port p and the exhaust e into the exhaust pipe
Ep, whence it escapes with a puff up the stack. The valve then
automatically connects p with the steam chest and p' with the
exhaust and the piston is again pushed backward, as in Fig. 98;
and so on. Thus the steam, by expanding in the cylinder, moves
the piston; and this motion is used
in doing the nTechanical work of
turning the drivers.

162. Work Done by the Steam.
The most important question about
a steam engine is: What is its Fio. 98. The^ piston Starts
efficiency? In order to answer

this question, we have to determine how much work is done by
the steam, and how much energy is supplied to it (cf. Art. 37).
How can we do this? How shall we measure the work done by
the steam? How determine the amount of heat energy supplied?

HEAT AND WORK 177

The work done by the steam may be determined with the help
of equation (5), Art. 34, W = fl. For if the total force of the
steam against the piston is /, and if this force acts through a dis-
tance / equal to the length of the stroke, the work done at each
stroke of the piston is simply the product of these two quanti-
ties. But the total force of the steam on the piston is the
steam pressure, i.e., the force per cm^, multiplied by the area
in cm^ of the piston. Therefore the work done by the steam is
W = steam pressure X piston area X length of stroke.

This result may be interpreted in another way. For piston
area X length of stroke is the volume of the cylinder; therefore,
the work done in one stroke is W = steam pressure X volume of
cylinder. Hence we can determine the work done by the steam
when we know these two factors.

In actual practice it is an easy matter to measure the volume
of the cylinder; but the pressure is not constant throughout the
stroke. When the steam enters the cylinder, it has the pressure
that exists in the boiler; and, if the port were left open during the
entire stroke, the steam would be exerting that same pressure
when the exhaust was opened. It would then expand suddenly
into the air, and thus waste a large amount of its energy. There-
fore it is customary to arrange the, valve V, Fig. 96, so that
it closes the port p when the piston has made about one-quarter
of its stroke. The steam then pushes the piston the remaining
three-quarters of the stroke by its own expansion. But while the
steam is expanding and doing this work, its pressure is decreas-
ing; therefore when the exhaust is opened, the steam pressure
is found to have fallen nearly to that of the atmosphere.

163. The Pressure- Volume Graph. Since the pressure of
the steam changes during the stroke, we must find an average
value to use in calculating the work. This is generally done by
measuring the pressure at different parts of the stroke by means
of an ingenious pressure gauge attached to the cylinder (c/.
problem 8, page 187). The pressures thus found are plotted as
ordinates with the corresponding volumes as abscissas, and the
average pressure obtained by measurements on the graph. Fig.

178 PHYSICS

99 is such a graph taken on a locomotive cylinder when it was

pulling a train. The point p^ represents the pres;sure and v^ the

volume when the steam had just entered the cylinder. As the

piston moved, the volume t\ increased, while the pressure remained

constant and equal to

fiQ, C^ that in the boiler until

the point pj was

reached. At this point

the port was closed and

the steam, being thus

"cut off" from the

boiler, was left to ex-

^^ ^^ pand. During the re-

Fio. 99. The Pressure-Volume Graph . , « .| . ,

mamder of the stroke,
while the steam was expanding and the volume increasing, the
pressure was falling as shown by the curve p2 p^. At p^, the
piston having now reached the end of its stroke, the exhaust
was opened, so that the pressure fell suddenly to that of the
atmosphere, represeifted by p^. As the piston returned, the
volume of the cylinder decreased; but because the exhaust was
open the pressure there remained constantly equal to atmos-
pheric pressure up to the point p^. At this point the valve
shifted so as to close the e^chaust and let in the live steam (c/.
Art. 161); and consequently the pressure jumped quickly to p^,
that of the boiler, and the entire process was repeated.

164. Back Pressure. In determining the average pressure
from a graph like this, one thing must be carefully noted. When
the piston returns, it has to push the exhaust steam out of the
cylinder against the pressure of the atmosphere. The work
of doing this is numerically equal to atmospheric pressure X
volume of cylinder. This amount of useless work must be sub-
tracted from that done by the steam, in order to find the amount
of available mechanical work done by it. Therefore, when an
engine exhausts into the atmosphere, as the locomotive does,
the average pressure by which its work is determined is the differ-
ence between the total average pressure and that of the atmosphere.

HEAT AND WORK 179

165. Lower Pressure at Exhaust. It has probably occurred
to many of you that we might increase the amount of work done
if we could allow the gteam to exhaust into a vacuum instead of
into the air; for then no work would have to be wasted in pushing
the used up steam out of the cylinder. This may be partially
accomplished by allowing the exhaust steam to escape into a coil
of pipes which contain no air, and which are surrounded by cold
water. When the exhaust steam enters these pipes, it is con-
densed into water, and so the pressure there is that of the saturated
vapor at the temperature of the cold water that surrounds the
pipes (cf. Art. 128 and Fig. 85). Such a device is called a
CONDENSER. The water condensed from the steam has to be
pumped out of the condenser against atmospheric pressure; but
since its volume is so much less than that of the steam, the useless
work done against the atmospheric pressure is much less. Con-
densers can not be used on a locomotive, because they are too
bulky, and because they require a large amount of cold water
to keep them cool. They are useful, however, on steamboats,
where space is more plentiful, and where a large supply of cool
water is always at hand.

166. Increased Boiler Pressure. Another way to increase
the amount of work done by the steam is to increase the boiler
pressure by increasing the temperature of the steam ; for the work
= average pressure X volume of cylinder, and this process in-
creases the average pressure. There is, however, a practical
liniit to increasing the efficiency in this way, because the pressure
must never be allow^ed to approach that at which the boiler would
burst {cf. Arts. 128, 129).

167v Heat Energy Consumed. Having now found how the
work done by the steam may be measured, let us consider how
the quantity of heat energy supplied to the engine is determined.
This is done practically by first noting the amount of fuel con-
sumed by the engine in a given time, and then finding out how
many heat units are liberated when this amount of fuel is burned.
For example, a locomotive of the type shown in Plate I con-

180 PHYSICS

sumes 1500 gm of coal for every horse-power that it furnishes
for one hour. The number of gm cal liberated by each gm of
coal when it is burned is found to be 7800. Hence the amount
of heat liberated by 1500 gm of coal is 1500 X 7800 = 117 X 10^
gm cal. Since the mechanical equivalent of 1 gm cal is 4.19 X
10^ ergs [c/. equation (9), Art. 154], this amount of heat is equal
to 117 X 10^ X 4,19 X 10^ = 49 X 10*' ergs. This is the energy
supplied to the engine for each horse-power hour.

Since (cf. Art. 43) 1 horse-power = 746 X 10^ — ^, and since
' - ^ sec -

1 hour = 3600 sec, 1 horse-power hour = 3600 X. 746 X 10^

= 268 X 10" ergs. This is the work done. The efficiency

is then ^ork done 268 X 10" - ^^ - q 055 - 5 5^^

"" ^^^""^ energy supplied ~ 49 X lO^''^ " 4900~ ^•"^^'" ^'^^''

168. Efficiency and Temperature. We have just found
that a locomotive, when considered from the point of view of
efficiency, is a very inefficient machine. Yet even stationary
and marine engines seldom have efficiencies greater than 17%.
This fact leads us to ask whether there is any theoretical
reason for this. Are the conditions under which every heat
engine must work such that its efficiency is necessarily small?
Or may we hope some day to make heat engines of high
efficiency?

We can find answers to these questions by carefully tracing
the heat through an engine. We note that heat is absorbed by
the water when it passes into steam. The steam then carries
this heat with it when it goes into the cylinder, where part of the
heat is used up in doing the work of moving the piston. The
rest of the heat is carried with the exhaust steam into the air or
condenser, and is then no longer available for doing work in the
engine. The essential things in this process are: 1. Heat en-
ergy is imparted to the steam at a high temperature (boiler tem-
perature); 2. The temperature of the steam must fall when work
is done; 3. Heat is given up to the condenser at a low temperature.

The only heat available for doing work is that given up by the
steam in cooling from the temperature of the boiler to that of the

Plate VI. A Triplk Expansion Pumping Engine,
Baden Station, St. Louis

HEAT AND WORK 181

condenser. Hence the heat energy available for work depends
on this difference in temperature. An engine that could convert
all of this available heat energy into work would be a perfect
engine.

The total heat energy in the steam when it leaves the boiler
depends in a similar way on how many degrees the boiler tem-
perature is above the absolute zero. We may then say that the
total amount of heat energy of the steam depends on its absolute
'temperature, i.e., on 273 + t (cf. Art. 123).

Finally, since the efficiency of an engine is defined as

heat converted into work ' . . . . . ., ,

— 7-r-n — I r-i — » and smce m a perfect engme the work

total heat supphed ^ ®

depends on the difference in temperature (t — t') between the

boiler and the condenser; and since the total heat energy sup-^

plied depends on 273° + t, we may conclude that the efficiency of

a perfect engine might be expressed by the ratio -^=^ — -.

169. Comparison of Efficiencies. Plate VI is a picture of
one of the large pumping engines that supply the city of St. Louis
with water. It is a triple expansion engine. The steam from
the boiler enters the smaller cylinder at the left of the picture and
there expands and cools somewhat. It then passes into the
second cylinder, in which it expands and cools some more; and
from this it exhausts into the third and largest cylinder, at the
right of the picture. From this it passes to the condenser. This
arrangement, by which the steam is allowed to do part of its
expanding in each of the three cylinders, has many practical
advantages, such as distributing the strains among three cranks
instead of concentrating them in one; greater compactness,
since a single cylinder that would do the work of the three would
have to be of enormous size; greater economy in steam, since the
fall in temperature in each cylinder is only one-third of the total
fall, so that the steam does not condense so readily into water in
the cylinder.

In this engine the steam has a pressure of 10 atmospheres
(760 cm of mercury) when it enters the first cylinder, and it ex-

182 PHYSICS

hausts into a condenser in which the pressure is 53 cm of mercury.
On consulting a table in which the relations between temperature
and pressure of saturated water vapor are given, we find that the
temperature corresponding to the boiler pressure is 180° C, and
that corresponding to the condenser pressure is 80° C. If
the engine were perfect, its efficiency would then be

<-r 180- 80 100^ ^
273+^273+180 453 ''''

It was found that 500 gm of coal was consumed at the boiler
for every horse-power hour furnished by the engine. Since the
locomotive just considered (Art. 167) consumed 1500 gm of coal
per horse-power hour, the efficiency of this engine is just three
times that of the locomotive, or 5.5 X 3 = 16.5%. The real
engine is. thus seen to be about three-quarters perfect.

Sometimes boiler pressures as high as 15 atmospheres are
used. The corresponding temperature at the boiler is found
from the table to be about 200° C. If the pressure in the con-
denser is reduced to 3 cm of mercury, the corresponding tempera-
ture would be 30° C. So the efficiency of a perfect.engine working be-

i~i' 200— 30 170
tween these temperatures would be 273^^^ = ^73 + 200 " 473 " ^^^^

nearly. Since boiler pressures greater than 15 atmospheres
are not very safe, and since it is very expensive to reduce the
condenser temperature below 30° C, 36% represents the prac-
tical limit for the efficiency of a perfect steam engine. No
steam engine has yet been made with an efficiency as high as
this.

170. The Gas Engine. The efficiency of a steam engine is
thus seen to be low because the range of temperature (t—f)
through which we can use the steam is comparatively small. In
the gas engine, Fig. 100, the conditions are more favorable. In
this machine the fuel is burned in the cylinder where the work is
done. The temperature of the gas in the cylinder may, therefore,
be very high. * A mixture of gas and air is introduced into the
cylinder and there exploded. The pressure developed by this
explosion pushes the piston outward. It is then pushed back

HEAT AND WORK

183

Fig. 100. A Gas Engine

by the atmospheric pressure: another explosion pushes it outward
again, and so on.

A good gas engine consumes about 16 cubic feet of gas from
the city mains for every
horse-power hour that it sup-
plies. The heat of combus-
tion of illuminating gas in
New York has been found to
be 18 X 10* gm cal per
cubic foot. Hence the heat
supplied by 16 feet of gas
is 16 X 18 X 10* = 288 X 10*
gm cal. This heat energy
is equal to 288 X 10'
X 4.19 X 10' ■■= 1200 X 10^'
ergs. This is the heat en-
ergy supplied to the en-
gine.

In Art. 167 the number

of ergs in a horse-power

U^,,« „, ^ f^„.wl 4^ U^ Fig. 101. The Old-Fashioned

hour was found to be Water -Wheel

184

PHYSICS

268 X 10^^ ergs. Therefore the efficiency of this gas engine is
268X10" 22%.

1200 X 10^'

171. Turbines. In the engines thus far considered the rotary
motion of the drivers or of the flywheel was produced by a trans-
latory motion of the piston to and fro. Engines of this type
are therefore called reciprocating engines. In all such engines,

Fig. 102. A Steam Turbine

useless work has to be done in starting and stopping the pis-
ton at each stroke; and this action alw^ays produces a jarring
which is harmful both to the engine and to the building or boat
in which it is placed.

The old-fashioned water-wheel, Fig. 101, and the modern water
turbines, such as are used so extensively at Niagara, illustrate
another method of converting kinetic energy of translation into
kinetic energy of rotation. The moving water is projected
against the paddles or blades of the wheel, and thus keeps it
steadily turning. Many attempts have been made to construct

HEAT AND WORK

185

an engine in which a wheel would be set into rotation by blowing
steam against blades or paddles on it. It is only within the
last few years that engineers have learned how to apply this
principle so as to make a steam' turbine equal in eflficiency to the
best reciprocating engines.

Fig. 102 is a picture of one of theSe modern steam turbines.
The cover has been removed so
that we can see how it is made.
Instead of a few large blades,
like the water wheel, it has many
rows of small blades fastened to
a steel cylinder called a rotor.
These movable blades pass be-
tween rows of stationary blades
fastened to the case of the ma-
chine. High pressure steam en-
ters the turbine at the smaller
end of the case, and, in elbowing its way amongst the forest
of blades, sets the rotor into rapid rotation. The arrangement
of the fixed and moving blades in this turbine is shown in Fig. 103.
The dotted line cemo indicates the path of the steam.

Turbines have now been so far perfected that their efficiencies
are greater than those of reciprocating engines. On account of
their freedom from jarring, their high efficiency, and their com-
pactness, they are now coming into general use. It is interesting
to note that the principle of the steam turbine was known to Hero
of Alexandria (b.c. 120). The technical difficulties involved
in the practical construction of a steam turbine of high efficiency
have delayed its perfection for 2000 years.

FIXED

MOVING

FIXED

p MOVING

Fig. 103. Blades in the Steam
Turbine/

SUMMARY

1. The mechanical equivalent of 1 gm cal is 4.19 X 10^ ergs.

2. A gas is heated when it is compressed and cools when it
does work in expanding.

3. When a liquid evaporates, heat is absorbed.

4. The work done by the steam in a steam engine is measured

186 PHYSICS

by the product of the average pressure and the volume of the
cylinder.

5. The efficiency of an engine maybe increased (a) by raising
the boiler temperature; (b) by using a condenser.

6. The efficiency of a perfect engine is equal to the difference
in temperature between the boiler and the condenser divided by
the absolute temperature of the boiler.

QUESTIONS

1. A tea-kettle of liquid air boils furiously when placed on a cake of
ice. Does this case differ essentially from that of a kettle of water on
a hot plate of iron?

2. Describe the experiment of Joule and Rowland. What relation
was established by them?

3. With the aid of diagrams made from memory, explain the action
of the steam in the cylinders of a steam engine, and the manner in
which this action is controlled by the slide valve.

4. Why is it possible to "ciit off" the entrance of the steam to the
cylinder before the completion of the stroke, and still get work out of
the steam that has entered?

* 5. What is the use of the condenser of a steam engine?

6. From the expression for efficiency in Art. 168, can you tell why
the cylinder of a gas or gasoline engine is arranged so as to be cooled
with water or air?

7. Why should the rotor and case of the steam turbine, Fig. 102, be
larger at the end where the steam leaves than it is at the end where it
enters?

8. We can convert a given quantity of mechanical energy into
heat. Can we convert a given quantity of heat entirely into mechanical
work? Why?

PROBLEMS

1. When a warm, moisture-laden wind strikes the sides of ^ moun-
tain range, it is forced up the inclined plane, and rises to where the
atmospheric pressure is less. What effect has this change of pressure
on its volume? Since it is expanding against some atmospheric pres-
sure, what effect does this expansion have on its temperature? What
effect may this change of temperature have on the invisible water
vapor that it contains? If the air then blows over the mountains,
will it be likely to deposit much rain on the other side?

2. How does Pascal's principle operate in the cylinder of an engine?
Does equa.tion (9), Art. 122, apply there?

HEAT AND WORK 187

3. Niagara Falls are about 53 m high. If all the energy of the fall-
ing water were transformed into heat, how much would each gram
heat itself by falling?

4. The cylinders of a locomotive are 60 cm long and 50 cm in diam-
eter. What is the volume of each? Take ir = 3.14. What volume
of steam is used per stroke? The average effective pressure of the
steam is found to be 3.5 X 10* dynes. How many ergs of work are
done per stroke? If 3.3 strokes are made per sec, what is the power
in -^^^? What is the horse-power? Remember that there are two
cylinders, and that a complete stroke is twice the length of the cylinder.

6. In June, 1892, in a test of the Empire State Express, the follow-
ing data were recorded when the train was going at a speed of 60 ~-^.
Find the horse-power developed. Area of each piston, 283.5 inches^;
pressure, 53.7 ^^^^^^^r^J length of cylinder, 2 ft.; revolutions per
minute of drivers, i.e., strokes of piston, 260.

6. Fig. 104 is a pressure-volume graph. When the volume is in-
creasing from pi to p2» does the pressure change? If the number of
cm in the length of the ordinate Vj pi be multiplied by the number of
cm in the length of V2 — Vi, what area does this product represent in
the figure? Does this area also represent the work done by the
gas in expanding from volume Vi to^ volume V2 ? Might we get the
numerical value of this work by multiplying this area of the rectangle
Vi pi P2 V2 by the number of dynes pressure and the number of cm'^
volume respectively, that 1 cm represents on the diagram? If the pres-
sure changed to a different value pa, for a change of
volume .V3— Vi, might we get in the same way
the work done during this new period of expan-
sion?

7. In the graph, Fig. 99, draw at equal dis-
tances ten vertical lines from the atmospheric
line p5 Pi to the broken line pi p2 Ps, measure
these linesj and take their average. Will this
average represent approximately the mean effec-
tive pressure of the steam against the piston? . If
we multiply this average by the length V2^^i» ^ ^z
what area will it represent? Will this area rep- f 104
resent approximately the work done by the

piston in traveling from one end of the cylinder to the other?
Why?-

8. Fig. 105 represents the pressure gauge mentioned in Art. 163.
The lower end of the tube at the left is coupled to one end of the engine
cylinder, and opens into it; so the steam can enter and push up the
little piston (seen inside the tube). This piston is held down by the
spiral spring near the top of the tube, but when the piston is pushed

188

PHYSICS

up against this spring, it lifts the lever. The lever carries a pencil
which it moves vertically along the little drum at the right of the dia-
gram. A card is wrapped around
this drum and held by the spring
clips. A cord, which is wrapped
around the drum, runs over a pulley
at the right of the drum, and thence
to a pin on the piston rod of the en-
gine. As the piston moves back, the
cord is unwound, and rotates the
drum. When the piston moves for-
ward, a spring turns the drum back
again and winds up the cord. Can
you apply your knowledge of the
composition of motions so as to tell
how the pencil draws on the card a
pressure-volume graph like Fig. 99?
This pressure gauge is known as the
steam indicator. It was invented by
James Watt. The graph is called by engineers an indicator diagram
and is much used in determining the work done by a steam engine.
The horse-power is obtained by dividing the work done per sec by 550
{cf. Art. 43), and is called the indicated horse-power.

9. What is the efficiency of an engine that consumes 1000 gm of
coal per horse-power hour?

10. An engine with an efficiency of 15 per cent does work at the rate
of 746 X 10* ^^. How many gm of coal must it burn per hour
{cf. Art. 167)?^^

11. In a water turbine, moving water turns the wheel and the
wheel turns machinery. Mention a case where a similar wheel is turned
by an engine and made to produce motion by pushing against the
water. A steam turbine and a windmill are used to propel machinery.
Mention a familiar case of a similar wheel that is turned by a motor
or engine and made to put air into motion.

Fig. 105

SUGGESTIONS TO STUDENTS

1. Hammer a nail rapidly for several minutes on a piece of iron,
pick it up and see what happens.

2. If you have a toy engine, bring it in and explain how it works.

3. Visit a roundhouse or engine shop and make a report of what
you see that interests you. Visit an ice and cold storage plant and
report.

4. Find out what you can about Count Rumford and his studies
of the conversion of work into heat.

HEAT AND WORK 189

5. You will find interesting information about heat in the follow-
ing books: Mach, Heat, Open Court Publishing Company, Chicago;
D. E. Jones, Heat, Light and Sound, Macmillan, New York; The Twen-
tieth Century Locomotive, Sinclair Company, New York. A good
description of the indicator diagram will be foimd . in PuUen's Me-
chanics^ pp. 253-260. You will find much interesting inforrfiation
concerning new forms of engines in the "Scientific American" and its
Supplement, in "The Engineering News," "The Technical World," Cas-
sell's and Cassier's magazines, and in "Railway and Locomotive En-
gineering."

6. Read the Life of James Watt, by Andrew Carnegie; the Life of
Robert Fulton, by R. H. Thurston. An interesting account of the de-
velopment of the steam engine is given in A History of the Growth of
the Steam Engine, by R. H. Thurston. Who was Count Rumford?
Read Memoir of Sir Benjamin Thompson, Count Rumford, by G. E.
Ellis (American Academy of Arts and Sciences, Boston).

7. The following books may interest you: Balfour Stewart, The Con-
servation of Energy (Appleton, N. Y.); Ray S. Baker, The Boys* Book
of Inventions, and the Second Boys* Book of Inventions (Doubleday-
Page, N. Y.); R. H. Thurston, Heat as a Form of Energy (Houghton-
Mifflin, Boston); T. O'Connor Sloane, Liquid Air and the Liquefaction
o/(?ases(N.W. Henley, N. Y.). .

8. A great deal.of useful information about all kinds of engines and
fuels is given in Wm. Kent, Mechanical Engineer's Pocket Book (Wiley,
N. Y., 6th Edition, 1903).

CHAPTER X
ELECTRICITY

172. The Transmission of Power. In the preceding chap-
ters, we have seen how the energy of wind, of water, and of steam
may be utilized in windmills, water-wheels, and steam engines
for doing mechanical work. When the energy of these contrivances
is transmitted by means of belts, cables, gear wheels, or shafting
to the machinery which it is to operate, this machinery and the
source of its energy must be near together. The eflBciency of
such a system of transmission is not so great as could be desired,
and it diminishes rapidly as the distance between is increased.
Furthermore, the practical difficulties in the way of these methods
of transmitting power become prohibitive at comparatively short
distances. When the work to be done is that of transportation,
the engine is a locomotive, and in doing its work it goes over any
distance desired; but it must carry with it a heavy load of fuel and
water, and it is not nearly so efficient for light loads and frequent
stops, as for heavy loads and without stops. Is there no form of
energy that may be transmitted cheaply and conveniently over
considerable distances, and used at such times, at such places,
and in such amounts as may suit the convenience of the user?

Most readers know in a general way that there are two meth-
ods by which energy may be thus distributed. One is by con-
verting the coal into fuel-gas, and sending it through pipes to the
various places where it is to be used for light, for heat, and for
operating gas engines for power. The other is by converting the
energy of the water-wheel or the steam eri^ne into that of an
electric current, and distributing it by means of copper wires to
electric lamps for light, and to electric motors for power.

173. Electric Generators. Plate VII is a photograph of one of
the large dynamo-electric machines of a power plant, whence
electrical energy 13 distributed. It is direct-connected with a

100

>
M

ELECTRICITY 191

big compound steam engine. The engine transforms heat energy
into mechanical energy; and the dynamo transforms the mechan-
ical energy into electrical energy. This electrical energy is sent
out along a system of wires to the places where it is to be used.
These powerful machines suggest many interesting questions
for study, some of which we shall try to answer in the next four
chapters. How do these machines work? What are some of the
elementary facts of electricity and of magnetism? What are the
relations between electricity and magnets? What are some of
the useful inventions by means of which the discoveries in elec-
trical science are applied so as to multiply both our means of
doing business and our facilities for enjoying Hfe? Who were
some of the great discoverers that sowed the seeds from which
this harvest has sprung?

174. Early Knowledge of Electricity and Magnetism. Be-
fore history began, man feared the lightning and thunder, and
had some primitive explanation for it. Amber, when rubbed
with wool, attracts light bodies. Doubtless the fact was known
long before Thales of Miletus recorded it, about 600 B.C. The
early Greeks also knew that lodestone, or magnetic ore of iron,
attracts pieces of iron; and some truth and many extravagant
fables were written of it by Pliny and others. The magnet is
supposed to have received its name from Magnesia, in Asia Minor,
where deposits of lodestone were found. The word "elec-
tricity" is derived from "electron," the Greek name for amber.

The Greeks never used the scientific method of study, and
hence they learned nothing of electricity beyond a few simple
facts, which in themselves are useless. They contented them-
selves with supposing that amber possessed a soul, which gave
it its strange powers.

176. Oilbert. The first man who ever studied electricity and
magnetism to any purpose was William Gilbert of Colchester
(1540-1603), a contemporary of Shakespeare and Bacon. Queen
Elizabeth appointed him her physician, and gave him a salary,
in order that he might be free to pursue his studies. He collected
and recorded all that was then known about the subject; and as

192 PHYSICS

a result of his experimental studies he discovered many new
facts. These he published in 1600 in a book, De Magnete, which
is still of good scientific repute.

176. Electrification. Gilbert found that other substances
besides amber, such as glass, sulphur, and the resins, would,
when rubbed, attract light bodies. When in this condition, they
are said to be electrified or charged with electricity. He
found also that he could not charge metals by rubbing them;
therefore he called the former electrics and the latter 7ion'
electrics.

177. Condnctors and Insnlators. Gilbert's lack of success
with his "non -electrics'' was due to a very important electrical
property, of which he failed to leam, but which was discovered a
century later by another Englishman, Stephen Gray. This
property, now so well known, is called conduction. An electric
charge will travel over or through some bodies very easily, some-
what as heat does; and it can thus be transmitted along them
from one place to another. Accordingly, these substances are
called GOOD conductors of electricity. Those substances which
do not conduct well are called poor conductors or insulators.
As a result of experiments, substances may be arranged as below,
in the order of their eleqtrical conductivities.

CONDUCTORS It is worthy of note that for most
Silver substances the order of their electrical
Copper conductivities is the same as that of
Iron their heat conductivities. There is no
Mercury dividing line between conductors and in-
Carbon sulators; for every substance has some
Solutions of Salts conducting power. The difference is
Pure Water merely one of degree.
Resins We ought now to know how to
Hard Rubber succeed where Gilbert failed. To elec-
Porcelain trify a piece of metal, we have only
Glass to fasten it to a handle made of one of
INSULATORS the substances near the foot of the list;
and we shall find that we can charge it as highly as we can any

ELECTRICITY 198

of the others. The human body and the earth are fairly good
conductors; and if it were not for the insulating handle of glass
or rubber, the charge would escape through the body of the ex-
perimenter, and spread itself over the earth. Consequently,
there would be so little energy left at any one place that no per-
ceptible work would be done by it.

178. Bepnlsion. That an electric charge may cause repul-
sion as well as attraction was first noticed by Guericke, who
devised the first electrical machine as well as the first air pump.
When two pith balls suspended by threads from an insulating
support are approached by an electrified glass rod, the rod at-
tracts the balls to itself. If the balls touch the rod, some of the
charge from the rod is communicated to them, and they are re-
pelled. The balls now repel each other; they also repel the rod,
as we should expect from the third law of motion (c/. Art. 40).
This may easily be shown by suspending the rod at its middle by
a paper sling, attached to a silk thread. In the repellent move-
ment, the rod, of course, has a smaller acceleration than have
the balls, because its mass is greater.

179. Discharge. If now the pith balls are touched by the hand,
or by any other conductor connecting them with the earth, their
charges will be dissipated. When this has occurred, they are
said to be discharged.

180. Two Kinds of Electrification. If the balls are both
charged from any other electrified body, they will repel each
other just as they did When charged from the glass. We find,
however, that their electrification is not always of the same sort;
for if we electrify them by contact with glass that has been rubbed
with silk, and then present to them a stick of sealing wax that
has been rubbed with flannel, we observe that though there is
repulsion between the balls and the glass, there is attraction
between the balls and the wax. Thoroughly discharge two pairs
of pith balls, A and B, electrify the pair A from the glass, and
the pair B from the wax. Those of the pair A repel each other;

194 PHYSICS

those of the pair B repel each other. But those of the pair A
attract those of the pair B, and are attracted by them. It was
thus found that there are two kinds • of electric charges, and it
became necessary to name them. That kind of charge which is
developed on glass by rubbing it with silk, is called a vitreous
or + (positive) charge; and that kind which is developed on
sealing wax by rubbing it with flannel, is called a resinous or —
(negative) charge. These names are, of course, purely arbi-
trary, and are adopted solely for convenience.

181. Bnfay's First Law. The results of experiments like
those just described are included in the following general state-
ment, which we may call the law of electrostatic attractions and
repulsions, or the first law of electrostatics. It is also known by
the name of its discoverer, Dufay.

Like charges repel each other; unlike charges attract each other.

182. To Determine the Kind of Charge. Since a charged
body always attracts an unelectrified body, as well as one having
a charge of opposite sign, attraction is not a satisfactory test of the
kind of charge that a body has; but if repulsion occurs between
two bodies we may be sure that they have like charges. If, then,
we wish to determine the sign of an unknown charge, * we may
impart some of this charge to a pith ball, and place the ball first
near a glass rod rubbed with silk, then near a stick of sealing wax
rubbed with flannel. If the ball is repelled by the glass rod, the--
unknown charge is of the positive kind; and if it is repelled by the
wax, the unknown charge is of the negative kind.

183. Electroscope. A suspended and insulated pith ball
thus serves as an electroscope, by means of which we may detect
a charge, and determine its sign; but for many experiments a
more sensitive instrument is needed. The one shown in Fig. 106
answers well. Two strips of gold or aluminum leaf take the
place of the pith balls; and the flask serves for an insulating sup-
port, as well as to protect the light and fragile leaves from any
disturbing currents of air. A very slight charge communicated

ELECTRICITY

195

to the metallic ball or plate at the top is conducted to the leaves,
and causes them to repel each other. Also, the greater the charge,
the greater the divergence of the leaves. When the ball or plate
is touched by the hand, the leaves collapse, showing that the
electroscope is discharged.

To test a charge by means of the gold leaf electroscope, give
the leaves a known charge suflBcient to cause a moderate diver-
gence. If now a charge of
the same sign is approached,
the divergence of the leaves is
seen to increase ; but if a charge
of the opposite sign is ap-
proached, their divergence is
seen to diminish.' Therefore, if
we have given the electroscope,
say, a negative charge, and if
we then bring near it the un-
known charge, an observed in-
crease in .the divergence of the
leaves will prove the unknown
charge to be negative, and a decrease in their divergence will
prove this charge to be positive. The proof plane P, Fig.
106, is a disc of metal with an insulating handle. It is used to
carry a small charge from a charged body to the electroscope
in order to test the body's electrical condition.

184. Both Snbstances Charged. We may now ask, Is it
likely that the substance with which we rub the glass or the wax
suffers no change of condition? Ought we not to suspect that it
also receives a charge? And will the charge, if it has one, be of
the same kind as that of the substance rubbed, or of the opposite
kind? For answer, rub the glass and silk together. When
tested with the electroscope, the silk will prove to be negatively
electrified. Rub the wax and flannel together; and the flannel,
when tested, will prove to be positively electrified. In this ex-
periment, the silk and the flannel must be tied to insulating han-
dles of glass or rubber, as they themselves are not sufficiently.

FiQ. 106. The Electroscope

196

PHYSICS

good insulators to retain their charges when held in the hand.
In this way it has been shown that whenever two dissimilar sub-
stances are rubbed together, one gets a positive charge and the
other a negative charge.

186. The Two Charges are Eqnal. But what about the rela-
tive amounts of the electric charges of the glass and the silk, or
the flannel and the wool? Are they equal or unequal? We may

call them equal if they pro-
duce equal effects, or if one
exactly neutralizes the effect
of the other. Let us make an
experiment which, though
rather crude, is nevertheless
convincing. Two brass discs,
A and 2?, Fig. 107, are fas-
tened to insulating handles.
The one on the right, in the
picture, is faced with a disc of
flannel or fur, of exactly its
own size, and neatly ce-
mented on with sealing wax.
First thoroughly discharge both discs and the electroscope.
Fit the discs accurately together, face to face, and twist one of them
half way around and back again, so as to rub them together.
Hold A two or three centimeters from the electroscope, and note
the amount of divergence of the leaves. Withdraw A, and put
B as nearly as possible in the same place. The leaves are seen
to diverge; and the divergence is the same in amount as before.
Now, without having allowed the discs A and B to touch
anything, fit them accurately together again; and while they
are thus held, bring them near the electroscope. Observe
that while they are together there is no effect on the leaves
of the electroscope, i.e., the two opposite charges exactly neu-
tralize each other's effects. Thus we know that they are equal
in amount.

These matters have been thoroughly and accurately tested

Fig. 107. The Charges are Equal

ELECTRICITY 197

by many experiments, all of which go to prove the following
general statement:

Any two dissimilar substances when brought into intimate con-
tact and then separated, acquire equal electrostatic charges of opposite
sign.

186. Pranklin's Theory. One of the most noted discoverers
in the field of electrostatics in the eighteenth century was. our
own Benjamin Franklin (1706-1790). Franklin proposed a
theory, which in his own time was very widely accepted. He
supposed that all unexcited bodies have an indefinite supply of
electricity, which is of one kind only, and that charges are gen-
erated by one body getting some of this electricity from some
other body, so that the former has an excess of electricity, while
the latter has an equal deficiency. The body having the excess
was said to have a positive charge, and the other an equal negative
charge. If was Franklin who proposed the use of the positive
and negative signs as suggested by this theory.

This theory is very useful as a working hypothesis. With
some slight modifications as to ideas and terms, it is still com-
petent to describe all electrostatic phenomena, including some
very remarkable ones recently discovered.

187. Electrostatic Polarization. Let us now see how we
can use this theory as a working hypothesis for the description
of phenomena and the discovery of new facts. In experimenting
with the electroscope, we can hardly have failed to notice a re-
markable fact which requires explanation, namely, that the leaves
diverge widely whenever an electrified body approaches the instru-
ment. This divergence occurs although the electrified body does
not touch the electroscope, nor even approach it near enough
for a spark to pass. Using our hypothesis, we may say, in explan-
ation, that the + electrification of the glass rod, when brought near
the uncharged electroscope, causes a disturbance of the neutral
electrification of the electroscope; it repels positive (+) elec-
trification into the end farthest away, and an equal negative
(— ) charge remains at the nearer end. The leaves, there-

198

PHYSICS

fore, being both positively charged, repel each other. In this
way, any neutral body may be given a + charge at one end, and
a — charge at the other, both of these charges being equal in amount
to the charge that causes this redistribution. A body in this
condition is said to be electrostatically polarized. Fig. 108, a.
That a conductor is really in the condition just described, when
under the influence of a near-by charge, may easily be shown by
taking small portions of its charges on a very small proof plane (P,
Fig. 106), and testing them with the electroscope. We can thus
prove that the end nearest the influencing charge has a charge of
opposite sign, that the other end has a charge of the same sign as
that of the influencing body, and that the middle region is neutral.

188. Orounding the Repelled Charge. If the influencing
charge be removed from the neighborhood without our having
touched the electroscope, the latter returns to the neutral
condition, as is evidenced by the collapsing of the leaves. We
may say, then, that the opposite charges at its two ends have
united and neutralized each other. But if we touch the plate

+ +■».+ + 4. 4. i + + •!• +

)

X \

I \

* A-

b "'^^'^E c '^ d

Fio. 108. Charging by Induction
a. Polarized, b. Grounded, c. Bound charge, d. Charged.

with the hand while the influencing charge is held near, the theory
leads us to expect that the repelled + charge, in trying to get as
far as possible from the glass rod, will go to the earth through the
body, while the — charge will remain on the plate, bound by
its attraction for the + charge on the glass rod. That the re-
pelled charge does go from the leaves to the earth is indicated by

ELECTRICITY 199

the fact that as soon as we touch the electroscope plate so as to
connect it with the earth E, Fig. 108, the leaves collapse. Allow-
ing a charge thus to pass to the earth along a .conductor is often

called GROUNDING THE CHARGE, Fig. 108, b.

189. Charging by Influence. The theory now suggests an-
other step. When we have polarized the electroscope and grounded
the repelled + charge, suppose that we remove the earth connec-
tion before we remove the influencing charge. Will not the
electroscope then be left with a — charge? It would seem so, be-
cause breaking the earth connection would certainly prevent the
return of the + charge, while the — charge would be held by its
attraction for the influencing charge. Fig. 108, c. When this bound
charge is released by removing the influencing charge, it will
move freely over the electroscope and reveal its presence by
causing a divergence of the leaves, Fig. 108, d. That this excess of
electrification is really present and is of the negative sort, may
easily be shown by the fact that the leaves collapse when the posi-
tively charged glass rod is brought near, and increase their
divergence when approached by a negative charge. This is the
most convenient method of charging the electroscope. Any con-
ductor whatever may be charged in this way. This kind of
charge is called an induced charge, and the process by which
a conductor B is electrified by another body A without loss
by A of any part of its own charge, is called charging by
influence. Let us review the steps of the process. They are:

1. Bring the influencing charge A near' the neutral conductor
B; B becomes polarized. 2. Touch B with the hand, or with
any earth-connected conductor; the repelled charge, equal in
quantity to A, and the same in kind, is grounded. 3. Break the
earth connection; the grounded charge can not get back, and the
bound charge, equal to A and opposite in kind, remains. 4. Re-
move A; the bound induced charge is freed, and spreads over
the surface of B, That the induced charge is equal to the
influencing charge may be proved, if both bodies, A and B, are
good conductors, by allowing A to touch B, when the two charges
will unite and neutralize each other. That the two charges are

200 PHYSICS

opposite in kind may be proved by testing them with the electro-
scope.

190. A Static Charge Besides on the Outside of the Con-
dnctor. When a charge is communicated to a conductor, any
two portions of it will repel each other; and therefore every por-
tion of the charge will get as far away from every other portion
as it can. Thus we should expect to find a charge distributed
over the surface of the conductor, and not at all on the inside.
That this is true, may be proved by using a very small proof
plane (Art. 183), with which to take samples of the electrification
from the different parts of the conductor. Thus we may electrify
an insulated, hollow brass globe, or even a common tin cup placed
on a cake of resin. With the proof plane, we may then test all
places on the outside of the hollow conductor, carrying to the
electroscope the sample charges, which will reveal their presence by
the divergence of the leaves. But try as we may, provided the
proof plane is not allowed to come too near the edges of the open-
ing in the hollow conductor, we can not succeed in getting any
charge from the inside.

191. Conlomb's Law. The magnitude of the force between
two charged bodies was first determined in 1777 by Coulomb,
an eminent French engineer arid physicist. He measured this
force by balancing it against the torsion (twisting force) of a
fine wire. He found that when the distance between two charges
was made twice as great, the force was reduced to J its former
value; and when the distance was made four times as great, the
force was reduced to yV- Coulomb's experiments also proved
that if either charge was increased in amount, the force in-
creased in the same proportion. The facts established by such
experiments may be summarized in the following general state-
ment, which we may call Coulomb's law of electrostatic force,
or the third law of electrostatics.

The force between two electrostatic charges varies directly cw
the product of their quantities, and inversely as the square of the
distance between their centers. This law is strictly true only when

ELECTRICITY 201

the charges are situated on spherical conductors, which are very
small in proportion to the distance between them. This state-
ment furnishes us with an appropriate unit in which to measure
a charge, for we may define unit charge as thai charge which,
placed in air at a distance of 1 cm from an equal charge of like
sign, repels it with a force of 1 dyne.

192. The Leyden Experiment. In the year 1745 a discovery
was announced from Germany, and a few months later from
Leyden, in Holland, which caused experimenters to redouble
their activity. While experiments were being made in electrify-
ing water in a bottle held in one of his hands, Musschenbroek,
a renowned science teacher of Leyden, touched the wire by
which the charge was passing into the water, with his other hand.
He receiyed a muscular shock of extraordinary power. The news
of the experiment spread rapidly, astonishing all Europe, and it
was repeated everywhere, both in Europe and in America, with
dramatic effect. At the French court, birds and small animals
were killed by electricity; and 180 soldiers in line were given a shock
simultaneously. A line of Carthusian monks 900 feet long was
formed, and the dignified ecclesiastics were made to jump up all
together, by the discharge of the ''Leyden bottle." All this
partook rather more of the spectacular than of the scientific;
but it served a good purpose, first in arousing general interest
in scientific experiments, and second in providing physicists
with a new combination to investigate, and a means of collecting
greater charges than had previously been at their disposal.

193. Condensers. The essential parts of the Leyden apparatus
are: 1. Two conductors of large surface; 2, a thin layer of
some good insulating substance between the conductors. The
two conductors are called the coatings, and the insulating layer
is called a dielectric. Such an arrangement of two conductors
and a dielectric is called a condenser. It soon took the familiar
form of a glass jar, coated inside and out with tin foil, reaching
to within 5 or 10 cm of the mouth. The mouth is closed with an
insulating stopper, through which passes a brass rod, terminated

202

PHYSICS

above by a brass ball, and below by a chain which touches the
inner coating. This is called a leyden jar. »

194. To Oive a Condenser a Large Charge, one coating must
be connected with one of the knobs of an electric machine, and
the other coating with the other knob or with the earth. As the
machine is operated, and the electrical energy given out, this energy
is stored in the condenser. To discharge the condenser , the two
coatings must be joined by a conductor. When the conductor
has almost completed the circuit, the air between the knob of
the condenser and the end of the discharger breaks down, or is
punctured, and a spark passes, accompanied by a loud snapping
noise. This is known as a disruptive discharge. Such sparks
are shorter, thicker, hotter, noisier, and in every way more ener-
getic than those which the machine can give without the con-
denser.

195. How the Condenser Operates. Since each of the coatings
of the condenser is connected with one of the poles of the electric
machine, one coating becojnes charged
positively and the other negatively.
These two charges are not able to
neutralize each other's effects, because
of the dielectric between them. They,
however, strongly attract each other,
and hold each other bound on the oppo-
site surfaces of the dielectric. Thus
the charges cling to the two sides of the
dielectric. This fact was discovered by
Franklin, and may be easily proved by
means of a Leyden jar, whose inner
and outer coatings can be removed.
Such a jar is shown in Fig. 109. After
the jar has been charged, the two coatings are removed and
discharged. When they are replaced, the jar will be found to have
retained its charge, for a bright spark may be obtained from it.
Hence the charge remained bound on the two sides of the

Fig. 109. The Charge is in
THE Dielectric

ELECTRICITY 203

dielectric, even after the outer conducting coatings had been
removed.

196. The Dielectric is in a Strained Condition. It has been
found useful to conceive that when a condenser is charged, the
dielectric between the two charges is in a strained condition.
This idea was first suggested by Faraday. The discharge of
the jar, then, consists merely in the release of the strain. Fara-
day also enlarged our conceptions of condensers by showing that
whenever a conductor is charged, an equal, opposite charge is
induced on some neighboring conductor, or on the walls of the
room. The dielectric between these two charges is in a state of
strain. Hence every charge, from that of a little pith ball to that of
a thunder cloud, may be regarded as the charge of a condenser;
for there are always two equally and oppositely charged conductors
separated by a dielectric.

197. A Disruptive Discharge is Oscillatory. Since we have
seen that an electric charge always implies electric strains in the
medium between the two oppositely charged bodies, we might
suspect that a discharge consists in the release of this strain.
We may get a rough mechanical picture of the state of things by
considering two plums, imbedded in a mass of elastic gelatin.
If the two plums are separated by stretching the gelatin, they tend
to come together again and resume the positions which they
had before the gelatin between them was strained. If we release
them by allowing them to slip back gradually, they will cease to
move when they reach the position of no strain. If, instead of
releasing them gradually, we release them suddenly, they first fly
beyond their positions of equilibrium, and then fly back nearly
to their starting points. So they continue to swing, or oscillate,
back and forth, but through smaller and smaller distances, until
they come to rest finally in their normal positions. Now, if an
electric discharge is the releasing of a strain in an elastic medium,
as we have conjectured, we can see from the consideration of the
crude gelatin model that when the spark passes, an oscillation
of some sort must take place. These electric displacements.

204

PHYSICS

first in one direction and then in the opposite direction, may be
conceived to be electric charges of opposite sign; and a disruptive
discharge would then consist of a rapid surging movement, or
alternating current, between the two conductors. If Faraday's
elastic displacement theory is anywhere near the truth, and if we
have reasoned correctly from it, we ought to conclude that the
spark from an electrostatic machine, or from a condenser, is
oscillatory, and it ought to.be possible to prove it by actual ex-
periment. It has been shown mathematically that the oscillations
which make up the discharge «,re exceedingly rapid. Neverthe-

FiQ. 110. The Spark Oscillates

less, it has been possible to photograph them and determine their
period, by means of a rapidly vibrating or rotating mirror. The
periods of • oscillation of the sparks from Ley den jars depend
mainly on the sizes of the jars, and range from one thousandth
to one ten-millionth of a second. Fig. 110 is a photograph of
such an oscillatory discharge. The mirror was turned to the left
while the charge surged alternately up and down between two
brass balls. The mirror, as it passed, threw an image of each
surging on to a photographic plate and since the mirror turned a
little each time, the successive images were thrown on different
parts of the plate.

198. Lightning. During his extended experiments with con-
densers and their powerful effects, Franklin began to be impressed
with the many resemblances between condenser discharges and
lightning. Conjectures of this kind had been advanced from
time to time by European physicists, from the days of Gray, but
in the mind of Franklin it had grown into a conviction. In 1749
he stated his conviction, gave his reasons for it, and proposed an

ELECTRICITY

205

Fio. 111. The Electric Spark in the Labo-
ratory

experiment with a pointed aerial wire by means of which the
electricity might be conducted quietly from the clouds to the
earth. His proposition
was received with skep-
ticism or indifference,
except at the French
court, where one exper-
imenter tried it and
was successful.

Franklin, hotvever,
did not regard this experiment as conclusive, because the
wire did not reach into the clouds; and he therefore thought
that the charge might have been received in some other way.
Accordingly, he devised an experiment which, for boldness of
conception and dramatic interest, as well as for its conclusive
character, stands unsurpassed in the history of electrical
research. He made a kite from a large silk handkerchief j and
tipped it with a pointed wire. He awaited a thunder storm,
and went out with his son to fly the kite. After sending the
kite up into the overhanging rain cloud, he held it by a strip

of silk attached to the
hemp twine, and waited
calmly for the appear-
ance of the sparks,
which he hoped to re-
ceive from a metal door-
key attached to the
string. " No man,' '
says a modem writer,
"ever so calmly, so
philosophically, staked
his life upon his faith.'*
He believed that the
pointed wire would
bring, not a disruptive
discharge, which he knew would kill him if it came, but a quiet
flow down the kite-string. At first there was no result; but after

The Electric Spark in Nature

206 PHYSICS

it had begun to rain, and the string had become wet, the hempen
fibers began to bristle up, and sparks came in plenty from the
key. A Leyden jar was charged, and the charge was thoroughly
tested for all the properties of electricity. Thus were the light-
nings and the thunders of Nature's great laboratory identified
with the sparks and the snappings of the machine in the philoso-
pher's laboratory. Thus were the valuable researches of **the
many-sided Franklin" crowned by a great discovery. Its pub-
lication produced a profound sensation in Europe as well as in
America. The honors which his researches brought him were
well deserved, for he was the greatest experimental philosopher
of his time.

In closing this chapter, it may be remarked that the benefit
TO MANKIND resulting from the study of electrostatics, up to the
point that we have now reached, was almost wholly intellectual.
Electrostatic phenomena are almost entirely devoid of practical
applications; but the intellectual progress which resulted from
the experimental study of these phenomena prepared the way
for discoveries of the phenomena of electricity in motion,
and for the vast throng of important inventions which depend
upon them. Some foolishly "practical'' person once asked Dr.
Franklin what use might be made of the facts proved by some
of his experiments. The great philosopher replied pithily by
asking him. What is the use of a baby? The discovery of the
oscillatory character of the condenser discharge illustrates the
force of Franklin's answer to this much asked question. What
is the use of a scientific discovery? In the course of time,
Maxwell deduced from theory that the ether (cf. Art. 148) might
be the medium in which electrostatic strains and oscillations take
place; and that if so, the discharge of a condenser must
start waves in the ether. Hertz went to work to start
such waves, and detect them. He succeeded. Then followed
the discoveries of more sensitive detectors by Branley and Lodge,
and the great work of the inventors in wireless telegraphy,
which is now going on. Like a baby, a scientific discovery
may be small and uninteresting to many; but no one can tell
how important it may become.

ELECTRICITY 207

SUMMARY

1. Mechanical energy may be converted into electrical energy.
This electrical energy may be economically transferred to distant
places, and there reconverted into mechanical energy, heat or
light.

2. In order to understand how this is done, one must know
the elementary facts and laws of electrical phenomena.

3. Amber and sealing wax, when rubbed with wool or fur,
attract light bodies/ and exhibit other remarkable properties.
They are then said to be charged with electricity.

4. Electrification travels along some substances easily, but
along others with great difficulty. The former are called con-
ductors, the latter, insulators.

5. Both conductors and insulators are essential to the trans-
ference of electrical energy.

6. There are two kinds of electrification, positive and nega-
tive.

7. Like charges of electricity repel each other; unlike, attract.

8. Any two dissimilar substances, when brought together and
then separated, become equally and oppositely charged.

9. The theory of Franklin affords a convenient language for
the description of electrostatic phenomena.

10. A neutral body may be. electrostatically polarized, and
charged by influence. The induced charge is equal to the in-
ducing charge, and opposite in kind.

11. A static charge does not exist anywhere inside a closed
conductor; it is always found on the outside.

12. The electrostatic unit quantity of electricity is that quan-
tity which, placed in air at a distance of 1 cm from an equal quan-
tity of like sign, repels it with a force of 1 dyne.

13. The repulsive or attractive force between two charges
is directly proportional to the product of their quantities, and
inversely proportional to the square of the distance between them.

14. A condenser consists of two conducting plates with a thin
layer of dielectric between.

15. The charge of a conductor is in the dielectric, not in the
conductor.

208 PHYSICS

16. Franklin proved that lightning is a disruptive electrical
discharge from cloud to cloud, or from cloud to earth.

17. The dielectric between two charged conductors is in a con-
dition of strain.

18. Every disruptive discharge is a condenser discharge.

19. A disruptive discharge is oscillatory and starts ether waves.

QUESTIONS

1. What great advantage has electrical energy over other forms?

2. What is the most obvious property of a body that is charged
with electricity?

3. What are conductors and insulators? Why are both necessary
to electrical transmission?

4. Why do we say that there are two kinds of electrification? How
are the two kinds named? How may the presence and the kind of
charge be determined with the aid of the electroscope?

5. With the aid of Franklin's theory, explain how a conductor may
be polarized, and charged by influence. How do the induced, and
inducing charges compare with each other in kind and amount?

6. Define the electrostatic unit of quantity.

7. What are the essential parts of a condenser? How is it charged
and discharged? Describe the manner in which the charge is accumu-
lated, accounting for the large capacity, i.e., ability to hold a large
charge.

8. What is a dielectric? Does the charge of a condenser belong
to it or to the coatings? How may this fact be shown?

9. What is the physical condition of the dielectric between two
charged conductors?

10. Describe a disruptive discharge. Explain why every charged
body must be regarded as one. of the coatings of a condenser.

11. What is the peculiarity of a disruptive discharge?

12. How may the heat, light, and noise of a disruptive discharge be
accounted for?

PROBLEMS

1. Two unlike electrostatic charges of Q and Q' units, respectively,
are distant d cm from each other. Will they attract or repel each
other? Call their force in dynes /, and write the expression for its
amount.

2. A certain charge is placed at a distance of 10 cm from a-f- charge
of 250 units, and the two charges are found to repel each other with a
force of 10 dynes. What was the amount and sign of the unknown
charge?

ELECTRICITY 209

3. In Fig. 108, a, suppose the charge of the glass rod is 10 units, and
its distance from the plate of the electroscope is 2 cm. How many — units
are attracted to the plate? How many -|- units are repelled to the
leaves? Suppose this + charge to be 10 cm from the repelling charge.
With how many dynes force is it repelled? What is the amount and
direction of the resultant force between the rod and the electroscope?
Do your answers suggest a reason why a charged body attracts an
uncharged body?

4. Does the self-repulsive property of a charge suggest why it is
found that when a body with sharp points is electrified, a large pro-
portion of its charge is collected at the points? It is also found in
such cases that the charge escapes rapidly from the highly charged
points, with streams of air that flow away from each of the points,
sometimes with force enough to blow out a candle. From your knowl-
edge of electric attraction and repulsion and of electrifying by contact,
explain how such streams of air are maintained until the body is dis-
charged.

5. Does question 4 suggest the reason why bodies intended to hold
electrostatic charges are usually made round and smooth? Does it
suggest why an electrostatic machine usually refusfes to "spark" when
a pointed wire is attached to it, or grounded and brought near it?
Does it show grounds for Franklin's belief that the charge from the
pointed wire on his kite would not harm him?

6. It is found that the electrostatic capacity of a condenser, i.e.,
its ability to accumulate a large charge under given conditions, is di-
rectly proportional to the area of its coatings, inversely proportional
to the thickness of the dielectric between the coatings, and also depends
on the material of the dielectric. If a charge is of the nature of a
strain in the dielectric, can you see a reason for each of these three
relations?

7. The dielectric of ^ condenser is often found to have a residual
charge, i.e., a second discharge is obtained from it when a little time
has elapsed after the first. Does this fact indicate that the charged
dielectric is in some such condition electrically as an elastic body is
mechanically when it is compressed, stretched, or twisted?

8. Can you show that an electrostatic charge has potential energy,
like a strained spring? What work is done to store this energy?

SUGGESTIONS TO STUDENTS

1. Read Tyndall's Elementary Lessons in Electricity (Appletons,
New York) for a fascinating account of electrostatic phenomena, with
many practical hints for those who wish to make experiments inex-
pensively for themselves. Read also Hopkins's Experimental Science^
pp. 359-391.

210 PHYSICS

2. Find out what you can about the early experimenters in the
field of electricity, especially Gilbert, Franklin, and Faraday. The
following books contain much that may interest you, and that you
can find easily if you will consult their indexes : Benjamin's The Intel-
lectvxil Rise of Electricity (Longmans, New York); Benjamin's The Age
of Electricity (Scribners, New York) ; Cajori's History of Physics (Mac-
millan, New York); Arabella B. Buckley's A Short History of Science,

If you have access to a good cyclopedia, consult it often for further
information about the men and the subjects mentioned in these pages.
The Encyclopedia Britannica is especially strong on the side of science.

3. If you will repeat for yourself the experiments mentioned in
this chapter, and as many of those made by your teacher as you can,
you will find that by thus becoming somewhat of an independent ex-
perimenter, you will not only increase your ability to understand the
lessons of this course, but you will also get a great amount of pleasure
out of it, and acquire a kind of skill that may be of great service else-
where.

CHAPTER XI

MAGNETISM

199. Lodestone and the Compass. Having learned in the
last chapter some of the fundamental facts concerning electric
charges and their properties, we will now take the next step
toward finding out how the dynamo operates; and seek to dis-
cover what a magnet is, what an electric current is, and what
relations exist between electricity and magnetism.

It is interesting to note in the first place that the attraction
of a magnet for iron has been known from time immemorial;
for magnetic iron exists in nature in the form of the mineral called
magnetite, or lodestone. This mineral was known to the Egyp-
tians and Greeks long before the Christian era, for in their writ-
ings we find them speculating about its attraction for iron.

Besides the attraction of lodestone for iron, little was known
of magnetism, and no use was made of it, until the introduction
of the compass in Europe by the Arabs, about the year 1200
A.D. A COMPASS, as is well known, consists of a small strip of
steel, which is first magnetized by rubbing it from end to end
on a lodestone, or any strong magnet, and is then suspended on
a pivot so that it is free to turn in a horizontal plane. Such
a magnetic needle always tends to set its length in a north
and south direction, and therefore it has been used by all
civilized people for determining the north and south line. Since
every magnet, when freely suspended, like the compass needle,
settles in a definite position, which is nearly north-south, the
magnet is said to have polarity. The end which points toward
the north is called its north-seeking pole, and the other end, its

SOUTH-SEEKING POLE.

The reason for this property of the magnet was not known
until the time of Gilbert. He conceived that the earth is itself a
great magnet, having its magnetic poles near the geographical

211

212

PHYSICS

'poles. To demonstrate that his theory was sound, he made a
sphere of a piece of lodestone, and showed that a small magnetic

needle, when near the surface of
this miniature model of the earth,
points in directions similar to
those in which it points at cor-
responding places on the surface
of the earth (Fig. 112).

FiQ. 112.

Gilbert's Model of the

EARTil

200. Magnetic Curves. This
study of the direction in which
a small, freely-suspended magnet
comes to rest in the neighbor-
hood of a large one, is of great
interest and importance. The
experiment may be performed as Gilbert performed it, by placing
a small suspended magnet at various points near the large magnet,
and marking at each point the direction in which it comes to rest.
The same result may be accomplished more quickly by cov-
ering the magnet with a sheet of cardboard, and sifting fine iron
filings over it. When the cardboard is lightly tapped, each bit
of iron acts as a small compass needle, placing itself with its axis
in the same direction as would a compass needle. Figure 113 is
a photographic reproduction of the result. It will be noted
that the iron filings trace well defined curves. These curves
indicate the direction of the mag-
netic force at every point about
the magnet. Many of these
curves appear to begin at points
on the magnet near its end. If
the card were large enough to
show them all entire, they would
all appear to end at points on
the magnet and near its other
end. There are two points, near
the ends of the magnet, toward which the lines of force appear
to converge. They are called the poles. We also note that the

Fig. 113.

Magnetic Field of a
Bar Magnet

MAGNETISM

213

curves do not intersect each other. It is customary to think of
the lines represented by the filings, as passing through the magnet
and forming closed curves.

201. Magnetic Field. Now, the behavior of the iron filings
tells us that the space about the magnet is permeated with
magnetic forces, whose directions are indicated by the curves.
This space around the magnet is called a magnetic
FIELD. The lines traced by the filings are called lines of mag-
netic force, because they show the direction in which a free north-
seeking pole would move at any point in the field. This direction
that a north-seeking pole would take is often shown by an arrow
point.

Let us now consider the field of force produced by two mag-
nets. To do this, we place two magnets under the card, and
let the iron filings trace
the directions of the lines
of force as before. Fig.
114 shows the result
when the adjacent poles
of the magnets are of
opposite kinds. It will
be noted that the lines
of force about each
magnet are distorted, and that some of the magnetic lines appear
to pass through both magnets. Since experiment shows that
unlike magnetic poles always attract each other, we are led to
conclude that the magnetic force is a kind of tension along the
lines of force, as if these lines were elastic and were trying to
shorten themselves.

If we investigate the shapes of the lines when like poles are
placed near together, we obtain the curves shown in Fig. 115. In
this case it will be noted that, though the field of each magnet
is distorted by that of the other, yet none of the lines of either
magnet enter the other. Since experiment shows that like
magnetic poles always repel each other, and since, in the space
between the two magnets, the lines proceed parallel to each

Fig. 114. Unlike Poles

214

PHYSICS

Fig. 115. Like Poles

other and in the same direction, we are led to think of the lines

of force as repelling each other in a direction at right angles to

their lengths.

We shall therefore
adopt the hypothesis
that a magnet is, in
some way, able to pro-
duce a strain in the
medium about it, that
this strain has a definite
direction at every point,

and consists oj a tension in the direction of the lines of force, and

a repulsion at right angles to that direction.

202. Permeability. The diagrams with the iron filings
enable us to foi-m a picture of the attraction between a magnet
and a piece of iron. For when we place a piece of soft iron be-
tween two unlike magnetic poles, the shape of the field of force
is that shown in Fig. 116. It will be noted that the field is dis-
torted, and the lines of force are concentrated by the iron. Some
of the lines of force that pass from one into the other go
through the iron; and so we have attraction. The iron thus acts
like a magnet, and so, in fact, it is. Its magnetism is said to
be indv^ced. So we see that iron, when placed in a magnetic field,
becomes itself magnetic by induction.

Further, the fact that the lines are gathered in and led through
the iron shows that the
magnetic force acts more
easily through iron than
through the air. This
property of gathering in
and conducting the lines
of force is called per-
meability. It is of great
importance in connec-
tion with all kinds of apparatus in which magnetic force is
used. By placing iron in the gaps between magnetic poles, the

Fig. 116. Permeability

MAGNETISM

215

lines are kept from leaking out; and the force, instead of being
dissipated, is concentrated within a small space. The more
nearly we can approach to having a closed magnetic circuit
of iron, the more efficient the apparatus will be. This is clearly
shown by the fact that a horse-shoe magnet is much more
powerful if it is fitted with a soft iron bar, or armature, as shown
in Fig. 117.

203. The Earth's Magnetism. The study of the direction of
the lines of magnetic force about the earth is of great importance
to navigation, for it has been found that the direc-
tion of the earth's magnetic lines do not coincide
with the true north and south direction, and
that the deviation is different in different places.
This fact was discovered by Columbus on his
memorable voyage in 1492, and when it became
known to his sailors, their fear drove them almost
to mutiny. As the voyage progressed, it was
found that the needle did not always point in
the same direction, but varied from the direction
of the pole star by different amounts; i.e., the
magnetic meridian does not always coincide with
the geographic meridian.

The departure of the needle from a geo-
graphic meridian at any point is its declina-
tion. The declination of the needle and the
variation in its amount in different places are easily ex-
plained by assuming that the magnetic poles of the earth do not
coincide with the geographic poles, and that the needle points
toward the former, not toward the latter. At points east of the
magnetic pole, the declination is toward the west, and vice versa.
From a study of the declination, it has been found that the earth's
north magnetic pole is situated in Boothia Felix, near Hudson
Bay, in latitude about 70°.5 north, and longitude 97° west.

The position of the magnetic pole itself is not absolutely con-
stant, and therefore we have a variation in the declination at
different times in the same place.

Fig. 117. Closed
Magnetic Cir-
cuit.

216 PHYSICS

The positions of the magnetic meridians, as well as other facts
concerning the earth's magnetism, have been determined by
different governments, at great expense, because the indications
of the compass to mariners and surveyors would be very inac-
curate without the corrections made necessary by the magnetic
declinations.

204. The Unit Magnetic Pole. We may now inquire how we
can compare the strengths of two magnets. In order to do this,
we must adopt a magnetic unit. The definition of the unit pole
adopted for scientific work is the following: A unit magnetic
pole is a pole of mich strength that when it is phiced at a distance
of one centimeter from a like pole of equal strength, the two repel
each other with the force of one dyne,

205. Law of Magnetic Force. When the measurements are
made in the units just mentioned, it is found that the force with
which two magnetic poles a^t on ea^h other is equal, numerically, to
the jyroduxit of their strengths divided by the square of the distance
between them,

206. The Chief Characteristics of Magnets niay be summed
up as follows: 1. When freely suspended, a magnet takes a
definite position, with its axis nearly north-south; 2, like poles
repel each other, while unlike poles attract; 3, we conceive a
magnet to be surrounded by a field of force, made up of lines of
force whose directions are indicated by the curves traced with
iron filings; 4, there is tension along the lines of force and repul-
sion at right angles to them; 5, a unit magnetic pole is one of
such strength that when it is placed at a distance of 1 cm from
a like pole, the two repel each other with a force of 1 dyne ; 6, a
piece of iron or steel, placed in a strong magnetic field, becomes
a magnet by induction; 7, steel is harder to magnetize than soft
iron, but it retains its magnetism better; 8, when a magnet is
broken into pieces, each piece is found to be a magnet.

207. Electric Currents. At the beginning of our study of
electricity we were led to ask how dynamos and motors work

MAGNETISM 217

but in order to find this out we had first to learn something
of electricity in the static condition and some of the proper-
ties of magnetism. We shall not be prepared to understand
the operation of electric machinery until we have learned a
few facts about electricity in motion and the relation between
electric currents and magnets.

In a general way we are all familiar with electric currents,
for we know that they are used to operate the telegraph, the tele-
phone, the electric light, the trolley cars, and even to ring our
door bells. Electric currents were wholly unknown until the
beginning of the nineteenth century.

The first knowledge of current electricity was derived from the
researches of Alessandro Volta (1745-1828), who was pro-
fessor of physics at the University of Pavia. Italian medical
men had been much interested in investigating the effects of the
electric shock on animal and himian subjects; and Galvani,
professor of anatomy at Bologna, was experimenting with the
legs of a frog. He found that when he twisted together the ends
of two wires made of different metals, and then touched a muscle
and a nerve of a dead frog's leg with the free ends of the wires,
the muscle would contract convulsively, and the legs would kick
as if they had been brought to life. The greatest excitement
followed this discovery, and the most extravagant hopes were
entertained. It was thought that electricity possessed the prin-
ciple of life, and would cure all ills.

Volta had already done much experimenting in electrostatic
induction. He believed that the electricity which caused the
frog's legs to kick was generated at the contact of the two dis-
similar metals, and not in the frog's leg, as Galvani contended.
By a series of very interesting experiments he proved that the
charge could be obtained from two different metals immersed
in a liquid and that it did not originate in the frog's leg.

208. The Voltaic Cell. This discovery was announced in
the year 1800. While seeking still further to increase the electric
output of his apparatus, Volta invented the simple cell which
goes by his name, and wliich has been but slightly modified in

218 PHYSICS

the best forms of modem commercial cells. This cell consists
of a copper and a zinc plate, each terminated by a wire and placed
face to face, but not in contact, in a jar of dilute sulphuric acid.
By joining a large number of such cells in series, i.e., the copper
of the first to the zinc of the second, the copper of the second
to the zinc of the third, etc., effects of considerable power were
obtained. His electroscope showed that the wire attached to
the copper was positively charged, and that attached to the zinc
negatively charged.

In 1800, only a few weeks after Volta had written of his re-
searches to the Royal Society at London, two members -of that
society, Carlyle and Nicholson, were experimenting with Volta's
apparatus, and discovered that if the terminals were placed in
water containing a little sulphuric acid, the water would be de-
composed into its constituent gases, oxygen and hydrogen, and
would be so decomposed continuously. This decomposition of
a chemical substance by electricity is called electrolysis. We
shall learn more of voltaic cells and electrolysis in Chapter XIII.
It interests us just now, because in this way it was first found
that the voltaic battery can produce, and transfer along a wire,
a constant supply or current of electricity. Thus electricity
was at once transformed • from a subject of purely intellectual
research 4nto a powerful means of investigation in every branch
of natural science, and a source of energy whose practical uses
are so numerous and far-reaching that the boldest imaginings
of that time are surpassed by the realities of the present.

209. Electromag^etism. Now, although the method of produc-
ing continuous currents had been discovered, no one had been able
to prove that there was any relation between such currents and
magnetism. That such a relation exists had been suspected, and it
was discovered in the year 1816 by Hans Christian Oersted (1777-
1851), professor in the university at Copenhagen. Oersted had
given much study and thought to the voltaic battery and the
possibilities of proving the long suspected connection between the
electric and magnetic forces. In a moment of inspiration, while
lecturing before his class, the idea occurred to him of joining

MAGNETISM 219

the wires from a battery above a suspended magnetic needle,
the wire being parallel to the needle but not touching it. The
needle instantly turned on its axis, and set itself at right angles
to the wire. He reversed the current, and the needle turned in
the opposite direction. He had shown that an electric current
possesses magnetic properties, in that it can move a magnet. He in-
terposed metals, glass, and other materials between the current and
the magnet, but found that none of them prevented the action
of the current on the magnet. Later it was learned that iron
will screen the magnet from the effects of the current, though
none but magnetic substances, such as iron, have this effect.

Oersted's great discovery was published in 1820, twenty years
after Volta's. The new territory which it opened was immediately
occupied, and other discoveries quickly followed. To the un-
trained and unthinking mind, Oersted's discovery might seem of
little importance; but let us see what we may learn from it by the
scientific method of inquiry.

210. The Current Has a Magnetic Field. From the fact
that the needle always takes a definite position with reference
to the direction of the electric current,
we have a right to infer that the cur-
rent has a magnetic field — that it is, in
fact, a magnet.

In order to test this inference, let
us take a wire conveying a strong cur-
rent, and dip it into a box of iron filings. ^HAs^i^Ao'jfE^Tic^r^ELD^
The filings cling to it, and if carefully

examined, will be seen to cling to one another so as to form a
number of rings. They do not stand out radially from the wire
as they do from the poles of a magnet, but are like a lot of
curtain rings strung along the wire. Break the current, and they
fall off. The field is instantly destroyed. In order more con-
veniently to investigate the form and extent of the field, let us
pass the wire up through a small hole in a smooth board, and
then down through another hole, close the circuit, sprinkle
filings, and tap the board. The filings are seen to jump into

■

fm^^^;'

t.V.>J.^!i?>^.'

220 PHYSICS

concentric rings around the two portions of the wire (Fig. 118).
The lines of force, then, are circumferences of concentric circles,
whose planes are all perpendicular to the direction of the current.
In which direction will these forces cause a north-seeking
pole to move? If we place a pocket compass at various points
around the wire, and mark by short arrows the directions of the
needle at these points, using the arrow tips to denote the direction
of the north-seeking pole of the needle, we shall see at once, from
the two maps, that when we look along the wire in the direction in
which the current is going; i.e., from copper to zinc, the direc-
tion of every line of force is that in which the hands of a clock re-
volve. Where the current is coming up through the board, the
lines of force appear to circulate counter-clockwise, but that is
only because we are facing the current so that it is moving
toward us. If we were to get down under the board and look
upward, the lines would appear to be clockwise as before.

211. Test for the Direction of a Current. This rule enables
us to predict the direction which the needle will take when placed
near a current, and conversely to tell the direction of the current,
if unknown, by the direction which the needle takes. Thus, if we
explore the field around the wire and find that there is a deflection
of the north-seeking pole clockwise with reference to it, we know
that there is a current traversing the wire, and that it is flowing
away from us. Conversely, if the deflection appears counter-
clockwise to us, then the current must be' coming toward us.

212. How are Currents Belated to Magnets? Notice again
the field of the current-bearing loop. Look along the top
of the loop in the direction of the current. The positions taken
by a compass needle show that all the lines go out of the left-hand
face of the loop, and enter the right-hand face, curving around on
the outside of the loop. Referring back to the field of the bar
magnet (Fig. 113), does not this suggest that the right-hand face
of the loop is a north-seeking pole? Make another loop with a
smaller diameter and with the lines closer together, the field
smaller and stronger.

MAGNETISM

221

■'*"'■ '1, ■ \

. ^^

.:- • ' '^. '

/ -

■ ■■■ - X ■,

'wm

■ '^%

t^M^

Fig. 119.

Magnetic Field of
A Coil

Can we not increase the strength of the field by making a coil
of several loops, so as to get the lines of all the loops into the same
space? On trying this, we find the
needle and filings more strongly
affected, and the map. Fig. 119,
much more distinctly like that
which a magnet ought to give us,
if it were made short and very
thick compared to its length.

Follow up this suggestion by
making a new coil in the form
of an elongated helix, and more like the bar magnet in shape
(Fig. 120). The field of this helix looks almost exactly like that
of the bar magnet. Each little loop has its own small circular
lines, but inside and outside the helix they combine to form
strong resultant lines which are closed curves exactly like those
of the magnetized steel bar.

We now have a right to infer that our current-bearing coil,
or helix is a veritable magnet. Will it do the things that a magnet
does? We have seen that it strongly attracts iron, and that like
the whole coil every little loop does so, though less strongly.
But, if freely suspended, will it point north and south? If placed
near another suspended magnet or another current-bearing loop

or helix, will the poles mu-
tually attract? Are the effects
strongest at its poles? The
requisite experiments show
that in every case the answer
is "yes," a current-bearing

conductor is a magnet,

213. Permeability— Elec-
tromagnets. We found (Art.
202) that soft iron placed in
the field of a bar magnet,
gathers in the magnetic lines, i.e., concentrates the force. Ought
we not, then, to be able to increase greatly the force of the coil or

Fig. 120. Magnetic Field of a Helix

222

phVsics

FiQ. 121. An Iron Core Strength-
ens THE Field

the helix by putting soft iron into it? On trying the experiment, it
is found that the iron core adds immensely to the magnetic strength

of the coil or the helix. We have
thus been led to construct an elec-
tromagnet (Figs. 121, 122). Such a
combination has two great advan-
tages over a steel magnet: 1.
Since its strength is proportional
to the strength of the current, and
also to the number of turns of the
wire aroimd the coil, we can thus
make a very strong magnet. 2. We can magnetize it and de-
magnetize it at will. We may now ask, can it not be maide to
do Various kinds of mechanical work at some distance from the
point at which the electric circuit is opened and closed?

This question leads us into a field in which notable discoveries
were made by an eminent American, Joseph Henry (1799-1878).
Henry was a busy teacher in the Albany Academy, and after-
wards professor at Princeton, and Secretary of the Smithsonian
Institution at Washington. He was incessantly overworked, but
in spite of that fact he made
researches that brought him
high honors among, scientists
all over the world.

214. The Electromagnet
and the Telegraph. Pop-
ularly, Henry's- name is
scarcely known in connection

Fig. 122. Field of an Electromagnet

invention has been given by
the American public to Sam-
uel F. B. Morse. But what Morse did was simply to combine prin-
ciples and apparatus discovered by Henry and others, and make
the public believe in the possibilities of the electromagnetic tele-
graph. Important as were the services of Morse, the honor of the

MAGNETISM 223

invention belongs to Henry, who, like his great English friend Far-
aday, was content to make fundamental scientific discoveries and
leave their practical applications to others. Recognizing the pos-
sibilities offered by the electromagnet, physicists everywhere were
trying to construct an apparatus for signaling at a distance by
means of it, and had given it up, because the current became so
weak after traversing a few hundred feet of wire, that the magnet
would not move anything. A member of the Royal Society had
tried, and claimed that he had demonstrated the impossibility of
the scheme. Henry read this paper and to his mind it was a
challenge; so with characteristic American audacity he set out
to accomplish the impossible.

Henry discovered that if he insulated his wires by covering
them with silk, and then woimd many turns of fine wire on the
core as thread is wound on a spool, this long coil magnet
would work at great distances from the battery, even though the
current was very weak. Using the horse-shoe form of core, he
pivoted in front of it a lever, carrying a little soft iron bar
or armature. The lever terminated in a clapper, which would
strike a bell when the armature was attracted. Placing this
apparatus in circuit with a battery of many cells in series,
he was able to make the clapper strike the bell whenever he
closed the circuit, and fall back in obedience to the tension of a
spring whenever he destroyed the magnetic field by breaking the
circuit.

Thus, representing each of the letters of the alphabet by a
combination of bell strokes, a message could be spelled out. In
the instrument which Morse afterwards patented, the lever car-
ried a pencil which it pressed against a moving roll of paper when
the armature was attracted. By making short and long contacts,
dots and dashes were made on the paper strip; and in the
Morse Alphabet each letter was 'represented by a combination of
dots, dashes, or spaces, thus:

a g t m 1

224

PHYSICS

The original idea of the Morse recorder survives in the "tick-
er*' of the stock exchanges and brokerage offices to-day, but the
instrument now generally used for commercial telegraphy is the
speedy and more familiar *' sounder," Fig. 123. Each signal of

the sounder is begun
by the fever L clicking
against a stop P when
the armature A is
pulled down on closing
the electric circuit,
which passes around
the core of the magnet
M. It is ended by the
lever clicking against
another stop Q when
the circuit is opened, and the armature, released from the magnetic
attraction, is pulled back by a spring 8. Thus the differences
between the dots and the dashes are represented by differences
in the time intervals between the double clicks of the sounder.

Fig. 123. Telegraph Sounder

215. The Belay. Neither a bell nor a sounder will work
over lines many miles long, however, because the current, weak-
ened by the resistance of the long circuit of wire, can not pro-
duce enough energy to do the heavy work. Henry overcame
this difficulty by devising the relay. He found that a single
battery cell with large plates would produce great effects when
the circuit was of short, thick wire,, offering little resistance to
the current, and that powerful magnetic effects could be produced
with such a battery by using a few turns of thick wire around the
core.

Suppose we have such a battery and magnet at the receiving
station connected in a circuit by 'short, thick wires. This magnet
SM, Fig. 124, with a strong current from a local battery LB can
operate the sounder lever SL, Now let us place in the main
LINE CIRCUIT L a magnet RM wound with many turns of fine wire.
The main current, though exceedingly weak, can furnish enough
energy to this sensitive magnet, so that it can work a very

MAGNETISM

225

light armature A, Fig. 125, with its lever. Now, this light
lever of itself can neither do any printing nor make any noise,
but it may easily make or br^ak an electrical contact between
two little points CC, one of which projects from the lever and
the other from a fixed metal post.

Suppose we cut the wires of the local circuit, and join
one of the cut ends to the contact point C on the relay lever, and
the other to the fixed point C against which^it strikes, as in Fig. 124.
All we have to do now to work the big
sounder, by means of the key at the
distant station, is to send our weak
main line current L around the core
of the sensitive relay magnet RM, The
relay armature is instantly attracted
and the contact point on the light ar-
mature lever strikes the fixed contact
point C. This completes the local cir-
cuit, and lets the powerful local current
gq around the core of the sounder mag-
net. The sounder lever is drawn down
and makes a loud click. Open the
main line circuit; and the relay arma-
ture, in obedience to the tension of its
spring S, Fig. 125, flies back. This
separates the contact points CC, thus
opening the local circuit; and the
local current ceases to flow around the
sounder magnet. Instantly, in obedi-
ence to the tension of its spring, the
sounder lever flies back. Thus the powerful local current is released
or throttled at will by the operator at the distant sending station.

This was the principle of Henry's relay. It is something like
using the weak current to pull a hair trigger, and discharge a
big gun. We make a powerful source of energy do heavy work,
but we control it by a weak current from a distant source. Long
distance transmission of electric signals of any kind is commer-
cially a practical impossibility without the relay.

Fig. 124. Telegraph Diagram,
One Station

£26

PHYSICS

Fig. 125. Telegraph Relay

216. The Telegraph Key. The opening and closing of the
circuit is accompHshed by a key, Fig. 126 and K, Fig. 124, worked

by the thumb and first
C C

-^^^ two fingers of the ope-

rator. It is simply a
lever with a contact
point P attached to its
under side, and strik-
ing another . contact
point attached to the
metal base, but insu-
lated from it. In tel-
egraphic practice there is a key, relay, sounder, and local bat-
tery at each station; and when one key is worked, it operates all
the keys and their sounders simultaneously. Accordingly, each
station has its own signal letter or call by which the attention
of its operator may be attracted.

217. Grounding the Wires. Before practical telegraphy had
progressed very far, the fact that the earth conducts electricity
suggested the possibility of using the earth as part of a telegraphic
circuit. This idea is realized in the following manner: One of
the battery wires is joined to a metal plate or pipe G, Fig. 124,
buried in damp ground, and the farther end of the line wire is
grounded in the same manner. The current which has passed
from the other pole of the battery along the line wire and through
all the instruments, proceeds to the ground connection at the
farther end, and com-
pletes its circuit
through the earth.
It is somewhat as if
we had a pump on
one side of a lake,
which would lift
water and force it
through a pipe to a place on the other side of the lake, where
it might do work in turning water motors, and then return to

Fig. 126. Telegraph Key

MAGNETISM

227

the lake when its energy was exhausted. Just as in the case of
the water there is a current through the pipe, and a drift across
the lake, so in the case of the grounded electric current, there
is an electric current in the wire, and an electric drift back along
the ground. As the ground resistance to the electric drift is very
small, this arrangement not only saves copper, but also saves en-
ergy, as we shall see later on.

218. The Electric Call Bell. The electric call bells and buzz-
ers in common use work very much like a telegraph sounder,

except that the armature which carries the ^

bell clapper, or the reed which makes the
tone of the buzzer, is made to vibrate auto-
matically, as long as the current is con-
tinuously supplied to it by keeping a push
button depressed.

The construction and operation of the
bell will be understood from the diagram,
Fig. 127. The path of the current may be
traced by the arrows. When the button is
pressed, the circuit is closed, the armature
is attracted by the electromagnet, and the
clapp_er strikes the bell; but at the instant
when this happens, the contact breaks be-
tween the contact-screw C and the armature,
because the armature has been pulled away from the contact
screw. The breaking of the electric circuit at this point destroys
the magnetic field; and the armature, no longer attracted, is
pulled back by the elastic spring on which it is mounted. Thus
it touches the contact screw; and since the circuit is now again
closed, the operation is repeated. Thus the armature vibrates
automatically, receiving its periodic impulses from the periodic
magnetic field.

Fig. 127. The Electric
Bell

219. Galvanometers. Shortly after Oersted's discovery. Am-
pere suspended a delicate magnetic needle inside a coil of wire,
and used it for detecting the presence of a current and estimating

228

PHYSICS

its intensity. Such an instrument is called a galvanoscope or a
galvanometer. When the plane of the coil is placed in the mag-
netic meridian, the needle, directed by the earth's magnetism,
remains in the plane of the coil — ^i.e., with its axis north-south —

unless a current is passed around
the coil. In this case, the magnetic
field of the current exerts a moment
of force on the needle, tending ,to
pull it around so that it will lie
along the lines of force of the coil —
i.e., east-west; and the angle through
which the needle is turned from the
north-south direction depends upon
the strength or intensity of the cur-
rent.

Many different types of the galva-
nometer are now in use. The most
useful are: (1) The astatic, a gal-
vanometer which may be made so
sensitive that it can be used for re-
ceiving signals sent through the sub-
marine cables, (2) The tangent gal-
vanometer, commonly used in measuring currents of considerable
strength, and (3) The D'Arsonval galvanometer. This last form
we shall pause to examine, both because it is the best for all-
around use, and because it will assist us in making another im-
portant step in our inquiries.

In the other forms we have a fixed coil with a needle sus-
pended at its center. In the D'Arsonval (a simple form is shown
in Fig. 128) we have a large, fixed, U-shaped steel magnet, with
a small coil suspended between its poles. The current to be
tested enters at the binding post E-\-, passes into the coil through
a thin metallic ribbon R by which the coil is suspended, and passes
out below by a similar ribbon. When the current passes
through the coil, the latter becomes a magnet and tends to set
its lines of force in the same direction with those of the large
fixed magnet. The magnetic lines of the magnet and of the coil

Fig. 128. D'Arsonval Galva-
nometer

MAGNETISM 229

are represented by arrows. The deflections of the coil are read
by a pointer moving over a circular scale, or by the displace-
ment of a beam of light, reflected from a mirror attached to the
coil. With all galvanometers the deflection of the suspended coil
or magnetic needle is greater when the current is greater. The
exact relation between the deflection and the strength of the current
depends on the kind of galvanometer.

}. A Suggestive Experiment. Let us pass the current from
a single voltaic cell through the coil of D'ArsonvaFs combinatioh.
The coil, Fig. 128, hangs with its plane parallel to that of the
U-magnet. As soon as the current passes, one face of the coil
becomes a north-seeking pole and the other a south-seeking pole,
as is suggested by the short, curved arrows which represent two
of its lines of force. The like poles of the magnet and coil repel
each other and their unlike poles attract each other. So the coil
turns through a right angle, and stops with its north-seeking
face next the south-seeking pole of the magnet.

If we reverse the current in the coil, its magnetic poles will
be reversed also; so that each pole of the coil will be facing a like
pole of the magnet. Now, we know that like poles repel each
other and unlike poles attract; therefore we may infer that the
magnetic forces will cause the coil to face about, so that each of
its poles will be adjacent to an unlike pole of the magnet.
When we shift' the battery wires, so as to reverse the current
through the coil, we find that the coil turns through half a revo-
lution, just as we inferred.

With the exercise of the scientific imagination we may now
arrive with a single bound at the principle of one of the greatest
inventions of all time. Can we, by any device, modify this appa-
ratus, so that when the current is again reversed, so as to be in
its original direction, the coil will turn through the second half
of the circle, instead of going back on its path? In other words,
can we, by reversing the current at the end of a half turn, cause
the coil to rotate continuously, instead of merely vibrating back
and forth through a semicircle? If we can but do this, what
boundless possibilities wait on the labors of inventors! Electric

230 PHYSICS

motors, turning machinery miles away from the source of current
— electric power, distributed by wires everywhere and converted
into mechanical work in the factory, the street or even in the private
home — power in just the amount wanted, available at the instant
when it is wanted, and the expense stopped the instant the power
is not needed, by merely turning a switch; these are among the
achievements which the device suggested would make possible.

This was the dream that possessed the imaginations of scientists
at the beginning of the second quarter of the nineteenth century.
Before the middle of that century the necessary discoveries had
all been made. The principles were in the hands of the inventors,
and in another forty years thousands of motors were in successful
operation.

221. The Electric Motor. In our electrical studies thus
far we have learned some of the most important facts and
principles that were known when Faraday and Henry were
seeking to discover the principles of the electric motor. Suppose
that the motor exists only in our imaginations as it then did in
theirs, and returning to our magnet and little suspended coil, let
us see what we can discover.

Let us send the current into the coil and observe what happens.
The coil rotates through a right angle, and sets its faces opposite
the poles of the magnet; but it does not stop at the instant when
it reaches that position. On account of its inertia, it goes
beyond that position. The magnetic force and the torsional elas-
ticity (twisting force) of the suspending ribbon stop it and bring
it back. After a few oscillations it settles into the position just
mentioned, in which its lines of force coincide with those of the
magnet.

If we can manage to reverse the current just as it passes this
position, and if we can also free the coil from the torsion of the
suspending ribbon, then instead of oscillating and settling in the
definite position mentioned, it will go on around through half
a revolution more. But on account of its inertia, it can not stop
itself; and if we again reverse the current just at the right instant, it
will continue to rotate through another half turn. Thus it appears

MAGNETISM

231

that if we can reverse the current just at the end of each half turn,
and if we can also get rid of the twisting force of the suspension,
we may produce continuous rotation.

By a little practice in timing the reversals of the current, we
may easily make the coil of a galvanometer execute one or two
complete revolutions, for in this case it stops only when the sus-
pension ribbon gets twisted up. Evidently we must overcome
this mechanical difficulty. This may be done by inventing a
sliding contact.

Armature, Field Magnets, Collecting Eings, Brushes.
We can arrange such a sliding contact by fastening the coil to
a steel axis or shaft, whose ends turn freely in suitable bearings.
It will be better also to mount the shaft horizontally rather than
vertically (Fig. 129). We shall now call the coil an armature and
the U-magnet the field
MAGNET. Then we may
place near one end of the
axis a pair of metal rings,
and solder the ends of the
coiled wire to the rings.
These rings, R -f and
R —, we shall call the

COLLECTING RINGS. In

order to let the current in

at one of these collecting rings and out at the other, we attach the
battery wires to two light metal springs, B + and B — , which we will
call the BRUSHES, and hold these brushes so that one of them shall
rub lightly against each of the collecting rings. While making the
apparatus, there is something very important which we must
remember and provide against. The steel shaft, or axis, about
which the coil rotates, is a conductor; and if our collecting rings
were in metallic contact with it, the greater part of the current,
coming to the first ring, woiild take the short cut along the steel
shaft from this ring to the other, and thence back to the battery.
Only a very small fraction of it would go around the turns of the
coil. Therefore our collecting rings must be mounted on a sleeve

Fig. 129. The Coil is Fkee to Rotate

S32

PHYSICS

of hard rubber which will insulate them from the armature shaft,
and from each other.

Having ready this apparatus, we make the experiment — first
touching the brush B -\- to ring R +, and the brush, B — to
R— ; then reversing the brushes, so that the brush B + touches R —
and the brush B — touches JB-f. If we are skillful enough in
shifting the brushes at the right instant, we secure continuous
rotation. We have, then, experimentally demonstrated the possibil-
ity of the electric motor; but our apparatus is as yet very crude
and inefficient^ even for a toy. What we need now is either an
alternating current, or a device for making the coil itself reverse
the current from the battery. We shall not here consider
the first proposition, because when the invention had reached
this point nobody had even dreamed of producing an alternating
current; i.e., one that would flow first in one direction and then in
the other.

223. The Commutator. Since tKe necessary motion of shift-
ing is only relative, it makes no difference whether the brushes

reverse their positions
while the positions of the
collectors remain fixed,
or whether we attach the
brushes to a fixed insu-
lated support, and let
the ^ collectors reverse
their positions. How
can we do this?

A little imagination
and some careful thinking will help us to hit on a plan. Man-
ifestly, when the coil turns, one-half of the ring is passing you
during the first half of the rotation, and the other half of the ring
is passing you during the other half of the rotation. It is, then,
easy to solve the problem by discarding one of the rings, split-
ting the other ring parallel to the shaft XX, and attaching the
two ends of the coiled wire to the two separated halves of the
ring.

Fio. 130. The Motor Diagram

MAGNETISM 233

We must now fix the brushes so that they touch diametrically
opposite points of the split ring. This arrangement is shown in
Fig. 130, the shaft being broken away and the split ring (C +,C —)
being enlarged for the sake of clearness.

If the brushes are properiy placed, it will be seen that at the
instant when the armature comes into the position where its poles
(i.e., its flat faces) are nearest the attracting poles of the field
magnet, the half ring C + will shift from the brush B + to the
brush B—, and the half ring C— will shift from the brush B—
to the brush B +. The result is that the current through the
armature coil is reversed, and so. its poles are also reversed.
Therefore the armature poles are repelled by the poles of the £eld
magnet, and so the coil continues to rotate. Since a similar rever-
sal of the current will occur at the completion of every half turn
of the armature, the latter will continue to turn so long as the
necessary power is supplied.

This split ring device we call a commutator, because it re-
verses the current through the coil whenever it shifts the brush
contacts.

The problem, then, of providing a sliding contact, and a
device for automatically reversing the current through the arma-
ture, is satisfactorily solved in our imaginations. If now we have
the completed apparatus before us, and send into the armature a
current of sufficient power, we shall be rewarded for all our labors
by seeing the armature spin merrily round and round. The solu-
tion of the problem is now a reality.

224. From Toy to Practical Machine. We must not stop
at this point, however, for our motor is very weak, being barely
able to run itself, to say nothing of driving machinery; and, fur-
thermore, we already possess knowledge which we may apply
to increase its efficiency. We know that the turning force is that
of the two magnets, and we have learned some ways of making
these magnets stronger.

1. We know of the permeability of soft iron (Art. 202), so
the next step is obvious. It is to fill the coil with a soft iron
core.

234

PHYSICS

2. We know that an electromagnet will have a stronger field;
so we can further increase the strength of the field, by substituting

an electromagnet for the
steel niagnet as in Fig. 131.

3. We may make our
magnet shorter and thicker,
with large pole-pieces, shaped
so as to embrace the coil as
closely and completely as
possible without interfering
with its rotation (Fig. 131).
This fills the air-gaps as
nearly as may be with soft
iron, and gathers in the lines
of force; so that more of
them pass through the arma-
ture, instead of leaking away
where they will not be util-
By a little further modification in shape, we may make the
completely **iron-

FiG. 131. Motor Frame, Poles, and Field
Coils

ized.
field

clad," as, represented in
Fig. 132. This is one of
the modern forms and
leaves little to be desired
in the matter of packing
the lines of force into the
effective space, and pro-
tecting the machine from
anything that might get
into it.

4. We may greatly in-
crease the efiPectiveness of
the armature by winding
on the core another coil at
right angles to the first,
so that when one coil is turning into the least effective position, the
other is turning into the most effective position. The number of

Fig. 132. A Modern Motor

MAGNETISM

235

Fig. 133. Motor Armature, Commutator,
AND Shaft

coils may be increased to four, six, eight, and so on (Fig. 133),
till all the available space on the core is filled, but the commutator
must be then re-di\dded so as to have one segment for each
coil. Increasing the number of coils not only increases the
magnitude of the force,
but also makes it approx-
imately uniform in inten-
sity.

225. Winding the Field
Magnets. The current may
be passed by the brushes
through the armature coils and thence through the field coils
(series winding, Fig. 134), or it may diyide at the brushes, part
going through the field coils and the remainder through the
armature coils (shunt winding, Fig. 135), or both these methods of
winding may be used together (compound winding, Fig. 136).
There are many other ways employed in the construction of

modern motors, by
which their efficiency is
still further increased.
Figs. 131 to 133 are
photographs of one of
the best modem types
of direct current motor.

226. Ampere's The-
ory of Magnetism. A
theory of magnetism
which was proposed by
Ampere is generally
held at present, though
it has received consid-
erable extensions in order that it may describe some facts discov-
ered since Ampere's time.

If we break up a magnetized knitting needle, we find that each
part is a magnet, even to the smallest parts into which it may be

II *

I" 'f

Fig. 134. Four Pole Motor, Series Wound

236

PHYSICS

Fio. 135. Two-Pole Motor,
Shunt Wound

broken. There is no reason why we should believe that the
smallest particles of iron into which a magnet can be divided,
should fail to have their magnetic poles. Ampere, therefore, be-
lieved that the magnetic
properties were permanently
possessed by the molecules
of the iron. The amended
and extended hypothesis of Ampere is:

1. Every molecule of a magnetic
substance is a magnet, possessing poles,
and all the properties with which we
have become familiar in large magnets.

2. When the magnetic substance is
unmagnetized, the magnetic axes of the
molecules are turned in all possible di-
rections, so that their magnetic forfces neutralize one another.

3. When the lines of force of an external magnetic field pass
through the bar, the little molecular magnets tend to set them-
selves along the lines of the external field.

4. When the molecules are all as nearly parallel to the axis of
the bar as is possible, the bar is said to be magnetically saturated.

5. The perfect magnetization of a molecule may be ac-
counted for by supposing, either that an electric current is always

circulating around it in a plane per- - _

pendicular to its axis, or that it has
an electrostatic charge and is rapidly
spinning. In view of the experiment
described in the next article the latter
idea seems to have the advantage.

227. Magnetic Field of Moving
Charges. In order to find out whether
a rapidly moving charged particle has
a magnetic field, just as a current
has, Rowland devised the following experiment: A metal disc
was mounted so that it could be rotated about an axle perpen-
dicular to its plane. When this disc is charged electrostatically,

Fig. 136. Two-Pole Motor,
Compound Wound

MAGNETISM 237

and is also rotating rapidly, it is found that a compass needle
near the disc is deflected just as it would be if a current of elec-
tricity were flowing in the path -of the rotating charge. Thus we
learn that charged bodies in rapid motion froduce electromagnetic
effects, just as currents do; and therefore we are led to conceive
that a current may be simply a series of electrostatically charged
particles darting rapidly along a wire.

The numerical relations obtained from this experiment are very
suggestive. It was found that if a unit charge (Art. 191) moves
with a velocity of 3 X 10*° ~, the strength of the magnetic field
produced is the same as that of a unit magnetic pole (Art. 204).
We shall leam in Chapter XXII that both light and the electric
waves, predicted by Maxwell and detected by Hertz (Art. 198),
travel with this same velocity, 3 X 10*® ^. These facts suggest
the idea that there must be some intimate relation between elec-
tricity and light. In Chapter VIII we learned that heat and light
are related phenomena. Therefore we may reasonably ask whether
electricity, heat, and light may not be simply different manifesta-
tions of one and the same form of energy. This question will be
considered further in the later chapters.

228. Energy of a Magnetic System. A magnet can not do
work, unless mechanical work is done on it, or a current of
electricity gives up some of its energy to it.

Thus, if a magnet has potential energy, so that it can attract
a piece of iron to itself, work had to be done to store up this energy.
Either the iron had first to be pulled away from the magnet, or the
magnet had to be pulled away from some other magnet from
which it got its magnetism.

If an electromagnet does work in moving a piece of iron, some
of the current which energizes the magnet is used up in doing this
work. Thus, if we pass a current through an electric lamp and a
magnet coil, the lamp diminishes in brightness while the work is
being done by the magnet. If we replace the lamp by a gal-
vanometer, the deflection diminishes while the magnet is doing
its work.

This is only another particular case of the general law of the

238 PHYSICS

CONSERVATION OF ENERGY, which states that no energy can be
created or destroyed.

SUMMARY

1. Lodestone is a compound of oxygen and iron, and is a nat-
ural magnet.

2. A magnet attracts iron, and when freely suspended, takes a
definite position with its axis nearly north-south.

3. The magnetic properties of a magnet are strongest at its
ends, which are called its poles.

4. Like magnetic poles repel each other, and unlike poles
attract each other.

5. A magnet induces magnetic properties in every piece of
iron or other magnetic substance which is brought into its neigh-
borhood.

6. Every piece of a magnet is itself a magnet.

7. The poles of a magnet may be determined, if unknown,
either by suspending it freely, or by presenting its poles to those
of a magnet.

8. The space that surrounds a magnet is a field of magnetic
force.

9. A vector drawn tangent to a line of force shows the direc-
tion in which a free north-seeking pole is urged at the point of
tangency.

10. Lines of force are always closed curves which pass out-
side the magnet from its north-seeking to its south-seeking
pole, and inside the magnet from its south-seeking to its north-
seeking pole. These lines never intersect each other.

11. Magnetic forces act as if the lines of force were elastic
threads which tend to shorten themselves lengthwise and repel
each other side wise.

12. If soft iron is placed in a magnetic field, the lines of force
are gathered in and pass through it. This is due to permeability
of the iron.

13. The earth is a great magnet, having one of its magnetic
poles near Hudson Bay. This explains why a magnetic needle
points nearly north-south.

MAGNETISM 239

14. The declination of the needle is explained by the fact that
the earth's magnetic poles do not coincide with its geographical
poles.

15. Variations in declination at a given place are caused by
the small variations in the positions of the earth's magnetic poles.

16. A unit magnetic pole is one which, when distant 1 cm
from an equal like pole, repels it with a force of 1 dyne.

17. The force between two magnetic poles, expressed in dynes,
is equal to the product of their magnetic strengths divided by the
square of the distance between them.

18. By chemical action a voltaic cell can supply a continuous
current of electricity, which transfers energy along a conductor.

19. An electric current has a magnetic field, and can do me-
chanical work by moving a magneti

20. Only magnetic substances act as screens to cut off
magnetic force.

21. The magnetic lines of a current in a straight wire are
circumferences of circles, whose planes are perpendicular to the
direction of the current.

22. The direction of a line of force is always clockwise to an
observer looking in the direction in which the current is going.

23. The magnetic field of a current-bearing helix is similar
to that of a similarly shaped steel magnet.

24. This fact leads us to infer that a current-bearing helix will
always behave like a magnet, and this conclusion has been fully
verified by experiment.

25. Placing a soft iron core in a current-bearing helix increases
the strength of its field. Such a combination is an electromagnet.

26. Electromagnets are used in telegraph sounders and call
bells to send signals over short conducting lines, and in relays
to send them over long lines.

27. Galvanoscopes are used to detect currents and determine
their directions; galvanometers, to measure their intensities.

28. An electromagnet, mounted on an axis and placed between
the poles of another magnet, may be made to rotate continuously.
This is the principle of the electric motor.

29. The efficiency of an electric motor may be greatly increased

240 PHYSICS

by arranging the soft iron parts so as to form a closed magnetic
circuit, apd also by increasing the number of coils in the armature.

30. According to Ampere's theory of magnetism, every mole-
cule of a magnetic substance is supposed to be itself a magnet,
because it carries an electric charge and constantly spins on its
axis.

31. A unit electrostatic charge, moving with a velocity of
3 X 10*® ^, is equivalent to a unit current and therefore produces
a unit magnetic pole.

32. A magnetic system can do no work unless it has been sup-
plied with energy, either by mechanical work, or by using up
some of the energy of an electric current.

QUESTIONS

1. Describe a series of simple experimeuts for demonstrating the
properties of magnets.

2. Explain how you can prove whether a substance is strongly
magnetic or not. If it is a magnetic substance how can you prove that
it is or that it is not a magnet?

3. What is a magnetic field? What is a line of magnetic force?

4. Describe the methods of mapping a magnetic field.

5. Sketch the appearance of the magnetic field of a bar magnet;
of two like poles repelling each other and of two unlike poles attracting
each other.

6. Sketch the effect on the field of introducing soft iron into the
magnetic circuit. What name is given to this property? What are
the effects on the properties of the field?,

7. What is the magnetic meridian of a place?

8. What great principle was established by Oersted's discovery?

9. Describe experiments by means of which the magnetic field of
a current can be shown.

10. How may we determine the direction of a current by means
of a magnetic needle?

11. Sketch the field of a current-bearing loop. How is the mag-
netic strength affected by multiplying the number of turns of wire so
as to make a compact coil?

12. What general fact may be inferred by the resemblance between
the field of any closed electric circuit and that of a magnet?

13. Suggest a series of experiments by which the fact thus inferred
may be verified.

MAGNETISM 241

14. What is an electromagnet? What advantages has it over a
permanent magnet? . ^

15. Diagram an electric call bell, trace the current, and explain
its action.

16. What is the principle of the galvanometer? State its uses.

17. Diagram a D' Arson val galvanometer and explain its action.

18. Describe the modifications that will convert D'Arsonval's
combination into an apparatus producing continuous rotation.

19. What are some of the most important changes that must be
made in the details of this apparatus in order to make an efficient
electric motor?

20. Describe the experiment of the rotating charged disc and tell
what conclusions we may draw from it.

21. Show that a magnetic system can have energy and do work
■only when energy has been supplied to it.

PROBLEMS

1. Write the expression that represents the magnitude of the force
/ in dynes, between a north-seeking magnetic pole of strength s and a
south-seeking pole of strength s' placed d cm apart. Is this force
attractive or repulsive?

2. When two south-seeking magnetic poles of 2 and 3 units strength,
respectively, are placed 6 cm apart, with what force do they affect
each other? Is the force attractive or repulsive?

3. What is the strength of a magnetic pole which exerts an at-
traction of 1 ,000 dynes on another pole which is distant 20 cm and has
a strength of 25 units?

4. With the aid of outline diagrams, describe the construction
and operation of the telegraphic key, the sounder, and the relay.

5. Diagram a telegraphic circuit of two stations, without relays,
trace the current by arrows, and explain the operation of sending a
signal.

6. Diagram a complete telegraph circuit for two stations, with
ground connections, battery, keys, relays, local circuits, local batteries,
and sounders. Trace the main line circuit around With black-ink
arrows, and the local circuits with red-ink arrows. Explain the action
of all the instruments when a key is worked.

7. Remove the cover from the push button of your electric door
bell, examine the contact spring carefully to find out how it works,
and make a diagram explaining its action.

8. Diagram a door bell battery of two cells in series, as you will
find them connected (c/. Art. 218). To this diagram add one of the
bell, and connect the battery with the bell by a wire. Add to the dia-
gram a wire returning from the bell through a push button to the

242 PHYSICS

battery, so as to make a complete circuit when the button is pressed.
Trace the current through the circuit by means of arrows, and explain
the action.

9. To your bell diagram, add several push buttons, as they would
be used to ring the bell from different points of the house.

10. Make a diagram representing a bell circuit in which the current
from the battery divides among several bells and reunites in a single
wire which conducts it back to the battery. Put into the diagram
one push button by which all the bells may be rung together.

11. Diagram an arrangement in which five bells are placed in a row,
and a current may go from a battery through any one of the he\\»,
from thence through a push button in a distant room, and from the
push button to. a common return wire which conducts it to the battery.
This is the arrangement of the annunciator seen in the offices of
hotels. There is a push button and an indicator for each room.

SUGGESTIONS TO STUDENTS

1. With a ten-cent toy magnet, a few sewing needles, knitting nee-
dles, some bits of broken watch springs, which your jeweler will give
you, a few bits of cork, some sealing wax, and a small amount of in-
genuity, you may verify all the properties of magnets, mentioned in
Arts. 199-206.

2. If you have a little shop of your own with a few good tools,
you may easily make yourself a simple D* Arson val galvanometer,
like that shown hi Fig. 128, a working telegraph set, a small motor,
etc.

3. You will find much helpful information about making such
things in Hopkins's Experimental Science; Electric Toy Making^ by
T. O'Connor Sloan (Norman W. Henley & Co., New York), and in a
series of little handbooks of the Bubier Publishing Company, Lynn,
Mass.

4. Get the necessary information from an electrical supply store,
and take charge of the electric bells in your home, keeping the battery
in order and the bells in adjustment.

5. Examine the field windings of a toy motor. Is it shunt wound
or series wound? See if you can change the connections so as to
convert it from one style of winding to the other. A shunt winding
requires many turns of fine wire, a series winding few turns of coarser
wire.

6. Connect a toy motor with a battery and a galvanometer, and
note the deflection. Now hold the armature so it can not rotate, and
see if the deflection is greater. If the first deflection is too great
connect a shunt across the galvanometer terminals (c/. Art. 268, Chap-

MAGNETISM 243

ter XIII). Does the experiment show that the motor takes energy
from the current while running?

7. If there is a shop in your city where electrical apparatus, such
as motors, is built, or a store where such things are sold, visit it and
find out .what you can. Many interesting electromagnetic devices,
toy motors, etc., are sold even in small electrical supply stores, and
tradesmen are usually willing to explain them to any one who is inter-
ested. Make a short written report of what you learn.

8. What can you find out about trolley car motors, and how their
power is transmitted to the car wheels?

9. The Central Scientific Co., Chicago, sell the parts of a small motor,
ready to put together. If you can not make a motor entire you can
easily assemble one of these.

10. If you know of a new house, where wires for electric bells and
annunciators are being put in, go in and find out how the wires are
arranged.

11. Find out, if you can, what changes should be made in the con-
nections of a motor in order to make it turn in the opposite direction.

12. Read A Century of Electricity by Prof. T. C. Mendenhall
(Houghton, Mifflin & Co., Boston), a very attractively written book,
with much about the history of discovery.

CHAPTER XII

INDUCED CURRENTS

9. Sonrce of Current. In the preceding chapter we have
seen how the modem electric motor might have been developed
from principles that had all been discovered as long ago as 1825.
Why was it that sixty years elapsed before it really grew into a
practical machine, and came into general use? The answer is,
that it had to wait for its counterpart, the d3niamo electric ma-
chine. And why? Because no matter how efficient the motor
itself may be, it must get its energy from the electric current.
The only means then known of supplying electric currents were
the various forms of voltaic cells, all of which derive their energy
from the chemical combination of zinc with oxygen, just as the
steam engine gets its energy from the chemical combination of
the carbon and hydrogen of the fuel with the oxygen in the air.
But the cost of zinc is so much greater than that of coal, gas, or
oil, that it costs a great deal more to do mechanical work with a
motor that is run by burning zinc in a battery, than it does to ido
it with an engine that is run by burning coal or gas under a boiler.
Thus the motor is of little practical value unless we can generate
electric currents in large quantity and at reasonable expense.
Where shall we look for the solution of this problem?

230. Current and Magnetic Field. In our studies thus far
we have found that many physical processes are reversible. Thus
a current of air will turn a windmill and do mechanical work.
Conversely, if we do mechanical work in turning a windmill back-
wards, we can make it act as a rotary fan, and produce a wind for
ventilating purposes. Heat may be converted into mechanical
work. Conversely, mechanical work may produce heat. There-
fore the question naturally arises: Since a current generates a
magnetic field, can not a magnetic fisld he made to generate a cur-
rent?

244

INDUCED CURRENTS 245

231. Faraday's Discovery. The discovery of how a current
can be generated with the help of a magnet was made by Michael
Faraday (1791-1867) in 1831. We shall be able better to ap-
preciate this great discovery if we repeat some of Faraday's ex-
periments. In order to do this we shall need a coil S, of many
turns of fine wire, a bar magnet M, a couple of voltaic cells or
other source of steady current, a sensitive galvanometer G, and a
pocket compass. This apparatus. Fig. 137, differs in no essential
way fr^m that used by Faraday.

Before making experiments, let us see if our previous study will

Fio. 137. The Moving Magnet Generates a Current

enable us to foretell what results we may expect. We know
that if we pass the battery current through the coil S, one of its
ends will become a north-seeking magnetic pole, and the other a
south-seeking pole. Lines of force will emerge from the former
and enter the latter. Let us send the current from the battery
in such a direction through the coil S that its upper end repels
the north-seeking pole of the compass needle, and is therefore,
itself a north-seeking pole.

Now insert the galvanometer G into the circuit, and note
its deflection. Suppose it is toward the right. We then know

246 PHYSICS

that when the galvanometer is deflected to the right, the current
circulates in such a direction in the coil that its upper end is a
north-seeking pole. Therefore, when the current is passing
in the coil, it is able to do the work of pulling a south-seeking pole
into the coil. If the phenomenon is reversible, we may expect
that if there is no current flowing in the coil, and if we do the
mechanical work of pulling the south-seeking pole out of the coil,
we shall generate in the coil a current, and that it will flow, in the
same direction as the current from the battery flowed. «

232. Current Induced by a Moving Magnet. In order to
verify this conclusion, we must remove the battery from the cir-
cuit, .leaving the coil and galvanometer connected as before.
Thert place the magnet inside the coil with its south-seeking pole
down, and see that the ga,lvanometer is at rest in the zero posi-
tion. When the magnet is quickly pulled out, the galvanometer
gives a quick right-handed deflection. Qur prediction was correct.
While the magnet is moirfng, a current is pissing in the coil. Since
the deflection is right-handed, we knew tliat the current which we
induced in the coil must be in the same direction as the battery
current. That current made the upper end of the coil a north-
seeking pole. The induced current, therefore, does the ,same.

But since the upper end was a north-seeking pole, it tended
to pull the magnet in, i.e., to stop its motion; so the induced
current did what it could to oppose the motion by which it was
generated. This is exactly what we ought to expect; for have we
not learned long ago that a perpetual motion machine is impossible,
and that if we produce some energy, as we have just done in
generating this current, we must do some extra work in order to
produce it? The extra work that we do is that of overcoming
the magnetic attraction between the induced current and the induc-
ing magnet pole.

Faraday was much puzzled, at first, by the fact that the in-'
duced currents were but momentary, for in his time the principle
that the energy stored and the work done in storing it are always
exactly equivalent to each other, was not so well known. It
ought to be perfectly plain to us, however, that the induced

INDUCED CURRENTS • 247

current can last just as long as the work continues, and no
longer.

We have jusi learned that when a south-seeking pole is with-
drawn from the coil, the upper end of the coil becomes a north-
seeking pole, which thus opposes the motion. What will happen
if we push the south-seeking pole of the magnet back into
the coil? If a current is induced in the coil, and if the direction
of this current is such as to oppose the motion, it should make
the upper end of the coil a south-seeking pole. When we try
the experiment, we find that the galvanometer gives a quick
deflection — not to the right, but to the left. But since a deflection
to the right means that the upper end of the coil is a north-seeking
pole, this left-hand deflection tells us that this upper end was a
south-seeking pole, as we predicted.

If we reverse the mjignet a/nd push its north-seeking pole
into the coil, we get a right-hand deflection, indicating an induced
current which makes the upper end of the coil a north-seeking
pole, and which thus repels the approaching north-seeking magnet
pole. When we withdraw the north-seeking pole of the magnet
from the coil, we get a left-hand deflectiony indicating an induced
current, which makes the upper end of the coil a south-seeking
pole, and thus tries to attract the north-Peking magnet pole and
oppose the motion of withdrawing it.*

Hence we conclude that when a magnet pole is pushed into a
coil of wire, or withdrawn from it, a current is generated in thai
coil. This current lasts only while the motion lasts, and is always
in such a direction that its magnetic field opposes the motion,

233. The Number of Lines of Force is Changed. A little reflec-
tion will enable us to see clearly that by all the four motions which
we made with the magnet, we either pushed lines of force into the
coil or pulled them out. We may, therefore, often find it con-
venient to conceive that the induced current is generated by
changing the number of lines of force that pass in one or the other
direction through the closed conducting circuit.

W'e should always remember, however, that when we vary the
number of lines of force in any of these ways, we must expect to

248

PHYSICS

expend some energy; for if we could vary them without energy,
and thus induce a current, we should be able to design a suc-
cessful perpetual motion electrical machine — a 'thing which all
competent minds agree is impossible.

234. Currents Induced by Currents. In the last chapter we
learned that a coil through which a current was flowing had mag-
netic poles, just like a magnet. We may, therefore, expect that
if we move such a current-bearing coil either into or out of another
coil, we shall get effects precisely similar to those obtained by
moving the magnet. The apparatus for the experiment b shown

Fig. 138. The Moving Con^ Mat Generate a Current

in Fig. 138. When we pass the current through the coil P, and
then bring it quickly near the coil S, the galvanometer gives
deflections as before, and these deflections again indicate induced
currents, which in all cases oppose the changes that we make in
the lines of force.

We have learned, then, that a current in one closed circuit
can be made to induce a current in another closed circuit. The
first current is often called the primary or inducing current,
and the second the secondary or induced current.

INDUCED CURRENTS

249

235. Iron Core. Since we know that more lines of force can
be gathered into the primary coil P by placing soft iron therein,
a new question is suggested, What effect will this stronger field
have on the secondary circuit? Will it vary the directions or the
magnitudes of the induced currents? The answer ought to be
easily forthcoming, for we know that there will then be more
lines of force to move into the coil or out of it, but that their direc-
tions will be the same as before. Therefore it is fair to infer
that a greater number of lines, moved in the same time, will induce
a stronger current each time, but that its direction will be the
same as that induced by the primary current without the soft iron
core C Again the appeal to experiment confirms our predictions.
The greater deflections of the galvanometer indicate the presence
of greater secondary currents.

236. Currents Induced by Making and Breaking Circuits.
There remains still another thing to try. By changing more

Fig. 139. Currents are Induced When the Circuit is Closed or Opened

lines in the same time, we change the number of lines at a greater
time rate, i.e., more quickly. Is there any way in which we may
further increase this rate of change? Evidently we can da so by

250 PHY&ICS

moving the primary coil more quickly, and on trial we find that
the induced currents are still greater.

But we have not yet reached the limit of increasing the rapidity
with which the number of lines is changed. We can change
them from zero to maximum, or vice versa, in a very small frac-
tion of a second by placing the primary inside the secondary, and
simply making and breaking the primary circuit. We may
verify this deduction as we did all the others (Fig. 139).

237. The Laws of Induced Currents. We are now ready to
formulate the results obtained by this interesting series of ex-
periments. They are :

1. Whenever the number of lines of force that pass in a
given direction through a closed conducting circuit is changed, a
current is indujced in that circuit.

2. Other things being eqv^l, the magnitude of the induced
current is directly proportional to the rate at which the number
of lines of force is changed,

3. The direction of the induced current is xd'Ways such that
its magnetic force opposes the motion which produces it.

The third law is known as Lenz's law, from the name of the
Russian physicist, Heinrich Lenz, who first announced it. It
may here be mentioned that when a current is induced in a sec-
ondary coil by starting a current in the. primary, the reaction of
the secondary current stops the primary, if it ican not push it
away mechanically; and the current induced by stopping the
primary tends to keep the primary going. The reaction, therefore,
may be electrical as well as mechanical.

We are now in possession of all the principles necessary for
the invention of the dynamo, the induction coil, the alternating
current transformer, and the telephone; therefore we shall now
take up each of these inventions in turn.

238. The Dynamo Principle. In the last chapter we learned
about the construction and action of an electric motor, and saw
how it may convert electrical energy into mechanical work. In
the present chapter we have learned that this process is reversible,

INDUCED CURRENTS

251

Fig. 140. The Dynamo Diagram

and that we can convert mechanical energy into electrical
energy.

But how are the laws of induced currents applied in the con-
struction and operation of the dynamo? With the aid of Fig. 140,
let us try to find out. In the diagram, N and S represent the
two poles of the field
magnet, their lines of
force being indicated
by the long vertical ar-
rows. FFi represents
a single coil armature,
CC a split ring com-
mutator, and B-{- and
B— a, pair of brushes
attached to the termi-
nals of the external
circuit around which the current is to be sent. The shaft and
bearings (c/. Figs. 129 and 130) are omitted for the sake of clear-
ness.

The armature coil being in the position shown, the greatest
possible number of lines- of force pass into the upper face F. But
when the armature is turned through a quarter of a revolution in
the direction of the curved arrows (seen at the right of the dia-
gram), its plane will be vertical. The number of lines of force pass-
ing into the face F will have been reduced from the maximum to
zero. Therefore an induced current will circulate around the coil
in such a direction as to make this face F a south-seeking pole,
which, by its attraction, opposes the rotation.

This current, whose direction is shown by the short, horizontal
arrows, will charge brush B+ positively and 5— negatively, and
will flow onward from B+ around the external part of the circuit,
returning to the armature through brush B— and commutator seg-
ment C. When the armature has turned through the second
quarter revolution, its plane will again be horizontal, and it will
again embrace the maximum number of lines of force. But since
the lines now enter the face F^, and since pushing the lines into
the face F^ has the same effect as has withdrawing them from F,

252

PHYSICS

the resulting induced current will continue to flow around the
coil in the same direction as before. Therefore the brush B+ will

again be positively
charged.

During the third
and the fourth quar-
ter revolution the
lines are withdrawn
from the face F^ and
pushed into the face
F. Therefore the
induced current

around the coil is
reversed during these
two quarter turns.
But it does not re-
verse at the brushes,
for at the instant
when the third quar-
ter turn begins, the
commutator segments reverse their contacts with the brushes, and so
B+ continues to be charged positively, and B— negatively. There-
fore the current flowing around the
external circuit is in the same
direction throughout the rotation,
i.e., from B+ to B—. This is
the principle of the direct current
dynamo.

239, The Dynamo. The power
and efficiency of a dynamo are
increased by the means previously
described in the case of the mo-
tor. The field magnets are elec-
tromagnets, and instead of two poles there may be four or more.
They are designed so as to give as strong and dense a field as
possible (Fig. 141).

Fig. 141. Six-Pole Field

Fig. 142. Armature and Com-
mutator

INDUCED CURRENTS

253

The armature consists of many coils wound on a soft iroti
core. Not only must the armature be carefully balanced me-
chanically, but the distribution of the coils must be such that the
moments of the magnetic forces are also symmetrically balanced
about the axis; otherwise the rapidly rotating armature will wabble
like an ill-balanced flywheel (cf. Art. 91). Furthermore, the
coils must be wound in slots in the core, and strongly bound in
their places; for if they were not held firmly in the slots, the mag-
netic forces that tend to stop
their motion would combine
with the centrifugal force to
pull them out of their places
(cf. Art. 90). The insulation
of the coils should also be as
perfect as possible. The
iron between the slots also
serves to fill the air gaps be-
tween the coils, and conduct
the lines of force into the
space where they are most
elTcctive. Fig. 142 shows
the armature, with the com-
mutator on the left. ' In Fig.
143 the assembled machine
is shown with the commu-
tator, brushes, shaft and one
of the bearings on the
right. Fig. 144 shows how an armature core for a very large
dynamo is built up of thin slotted plates of soft iron.

240. Winding of the Field Magnets. In the direct current
dynamo the current generated by the armature is used to excite
the field magnets. Accordingly these are called self-excited,
to distinguish them from the alternating current machines, which
must be separately-excited by a direct current from a small
separate dynamo. In the direct current dynamo, as in the motor,
the armature current may be carried around the field coils from

Fig. 143. Complete Dynamo

254

PHYSICS

Fig. 144. Building an Armature Core

the positive brush, then to the external circuit, and thence back
to the negative brush (series winding, c/. Figs. 134 and 152); or

it may divide at the
brushes, one branch
going around the field
coils, and the other
around the external
circuit (shunt winding,
Figs. .135 and 153); or
both styles ' of winding
may be used together
(compound winding,
Fig. 136). Fig. 141
shows the long, shunt
coils, next the iron-clad frame and the short, series coils close
to the ends of the pole-pieces. It is therefore a compound wound
field.

241. How the Field of Force is Built up. As the field magnet
cores are of soft iron, it may be asked. Since the cores are not
magnetized unless the current is flowing around them, and since
a current can not be induced in the armature unless lines of force
from the field magnet pass through it, how i^ it that the current
can start at all? The answer is, that the field cores always
retain a little of their magnetism after having once been strongly
excited. This residual magnetism is sufficient to generate a
small induced current in the revolving armature, and this small
current, in turn, increases the magnetic strength of the field mag-
nets. They are then able to induce a still stronger current in
the armature. The magnetism of the field cores and the resulting
current in the armature thus add gradually each to the strength
of the other, until the field magnets are saturated.

242. Magnetos. The early dynamos were magnetos, i.e.,
their field magnets were permanent steel magnets, resembling
that in Fig. 140. Such machines are still much used in operating
call bells on private telephone lines, and in producing sparks for
the ignition of the gases in gas engines.

INDUCED CURRENTS

255

243. Alternating Gnrrent Dynamos. For many purposes,
an alternating current has very decided advantages over a direct
current. In general principle the ''alternator" resembles the
direct current machine, but it has collecting rings (c/. Fig.
128) instead of comniutator segments, so that the electric impulses
sent out to the line change direction every time a pair of its
coils passes a pair of its poles. As there are usually several pairs
of poles and as many pairs of coils, there will be several alterna-
tions at each revolution.

244. The Induction Coil. The induction coil (Fig. 145)
is an instrument frequently mentioned in the papers and maga-

FiG. 145. The Induction Coil

zines, because it is used in producing X-rays, and in starting the
ether waves by which messages are sent through space without
wires. It was invented by an American, Charles G. Page, in
1838. It consists of a primary coil P, of a few ttirns of coarse
wire, containing a soft iron core, and surrounded by a secondary
coil Sy of many turns of fine wire, whose ends lead to a pair of
insulated knobs or points TT, Thus far it is precisely like our
apparatus for investigating induced currents (Fig. 139).

%5e

PHYSICS

But for convenience and speed in making and breaking the
primary circuit, there is usually added an automatic contact
breaker Hy which keeps itself vibrating, and automatically opens
and closes the primary circuit exactly as the armature of the
electric call bell does (Art. 218), Alternating currents are thus
induced in the secondary coil.

Sincea current impulse or pressure, called electromotive force,
is started in every turn of the secondary coil, every time that the
primary circuit is made or broken, it follows that these impulses
in all the turns will be added together. Therefore, up to a certain
limit, the pressure of the induced current increases with the num-
ber of turns in the secondary coil. The induced electromotive
force is also proportional to the suddenness with which the pri-
mary current is started or stopped (c/. Art. 236). As the alter-
nating induced currents surge back and forth in the' secondary
coil, the electrical pressures at the terminals TT become so
great that disruptive discharges occur between them. A 40-inch
spark coil produces a pressure equal to that of from 00,000 to
iOO,000 voltaic cells and contains over 250 miles of wire in the
secondary coil. Large induction coils must be designed with
great care, especially with regard to the insulation, which would
otherwise be punctured by the great electrical pressures.

245. The Alternating Current Transformer. Fig. 146 rep-
resents the apparatus used by Faraday in one of his earliest ex-
periments with induced
currents. It will be
seen that when a cur-
rent is started in one of
\he coils P, it will send
its lines of force around
through the iron of the
ring C, and thut when
these lines of force enter
the other coil S, they will induce in it a current whose direction
is opposite to that of the primary and which may be detected
by the galvanometer G. If the primary current stops, a cur-

FiG. 146. Faraday's Ring

INDUCED CURRENTS

257

Fig. 147. A Trans-
former

rent is induced havlpg the same direction as the primary. If

the primary current is alternating instead of intermittent, its

lines of force will enter the secondary coil first

from one direction and then from the other

alternately, and will thus produce alternating

currents in the secondary. . Since the elec-
tromotive force or pressure in the secondary

coil is proportional to the number of turns

of wire in it, it follows that the electromotive

forces in the primary and secondary coils are

proportional to the corresponding numbers

of turns of wire in the two coils. For ex-
ample, if the primary coil has 100 turns and

the secondary 100,000, the electromotive force

of the induced current will be 1000 times as

great as that of the inducing current.

Thus we may send an alternating current

of low pressuie and large quantity into the

short coil of such a "transformer,'* and get

out of the long coil an alternating induced current at high

pressure, and of proportionally smaller quantity. Conversely, we

can send an alternating current of high

* M pressure into the long coil, and get out of

"' the short coil an alternating current of

lower pressure and proportionally larger
quantity. If used in the former way,
the apparatus is called a "step-up"
transformer; if in the latter way, a
"step-down" transformer. The induc-
tion coil is a "step-up" transfonner.
^^^^^^^^ Transformers are used extensively in

P?|8^^^^^B electric lighting and car service, because
"^^^H the current can be transmitted with far

greater economy at high pressure than
at low pressure. High pressure cur-
rents, however, arc not suitable for use in lamps, and are not
permissible in buildings, because of the danger of fire and loss of

Fig. 148. CoNSTRurTioN OP
THE Transformer

258 PHYSICS

life. High pressure currents from alternating dynamos are,
therefore, distributed to transformers similar in construction to
Faraday's ring. These are placed on poles outside the buildings,
Fig. 147, and the low pressure currents are carried into the
buildings, for service either in electric lamps or in alternating
current motors. Fig. 148 shows such a transformer with the
outside case taken off.

246, Alternating Current Motors. There are two kinds of
altej-nating current motors. Synchronous motors resemble
alternating current dynamos in construction. They are so called
because the alternating currents in their fields and armatures
keep time, or step, with those in the dynamo that funiishes the
current to them. The armatures of induction motors are turned
by the magnetic forces acting between the field currents and the
resulting induced currents in the armature. They require very
little care, because the armature currents* have no electrical con-
nection with the supply wires, and therefore they have no sliding
electric contacts.

The study of alternating currents and induction motors,
though exceedingly interesting, is beyond the scope of an ele-
mentary course.

247. The Telephone. How is it that sounds so complex
as those which are produced by spoken words, with all their vari-
ations of loudness, pitch, and quality of tone, can be taken up
by a small piece of sheet iron and transformed into electrical
waves? And how is it that these electrical waves, after traveling
along hundreds of miles of wire, can be retransformed into sound
that is a close copy of that produced by the voice of the speaker?
This is, indeed, the most marvelous of all the facts that our studies
in Physics have yet brought to our attention. And the more we
learn of sound, and the great complexity of the motions which it
impresses on the air, the more we shall be led to wonder, that a
pair of such simple contrivances as a telephone transmitter and
receiver can work such a miracle.

The construction of the telephone is easily understood. The
parts are shown in Fig. 149. First, there is a funnel-shaped

INDUCED CURRENTS

259

mouthpiece M, iivto which we talk. This mouthpiece keeps
the sound from spreading into space, and directs it against a
diaphragm or disc of sheet iron Z), placed just at the end of the
funnel. Behind the diaphragm D, and attached to it, is a smaller
disc of carbon E, and just behind £ is a second disc of carbon
E, which is attached at its back to a metallic plate B, The small
space between E and E is filled with grains of hard carbon.

The metal supports on which the carbon discs E are mounted
are insulated from each other; but one is electrically connected
with one of the terminals of a voltaic battery Ba, or other source
of current, while the other is connected with one of the terminals
of the primary wire of a small step-up induction coil I. The other

Fig. 149. Telephone Transmitter, Receiver, and Circuit

terminal of this primary wire is connected with the other ter-
minal of the battery. A current of electricity is thus always pass-
ing between the carbon discs £, across the loose contacts of the
carbon gi'anules, and completing its circuit back to the battery
by the way of the primary wire of the induction coil.

Now, a loose contact between two pieces of conducting matter
has the very remarkable property of conducting a current better
when the pressure on it increases. Very small variations in pres-
sure at the points of contact produce changes of considerable
magnitude, in the strength of a current of electricity that may be
passing across the contact. Such a loose electrical contact is called
a microphone contact because it enables us to hear very faint

260 PHYSICS

sounds. The two carbon discs with their microphone contact, the
'battery, and the induction coil, are the only essential parts of the
telephone transmitter; but the metallic case is of course necessary
to protect these parts and hold them in their places.

The receiving instrument, or Bell telephone, is even more
simple. Its only essential parts are a strong steel magnet if,
whose poles are surrounded by coils of fine wire CC, and a dia-
phragm D' of sheet iron like that of the transmitter. Even
this diaphragm is not absolutely necessary, for the sounds
can be heard without it. The instruments are connected, as
shown, the diagram for the other station being exactly like Fig.
149, except that it is reversed. Their action is partially explained
as follows:

1. The sound sets the diaphragm D of the transmitter
into vibration. 2. In some way not yet completely under-
stood, these vibrations, when communicated to the micro-
phone contacts, produce variations in the electric current flowing
through them. These variations are counterparts in every detail
of the sound vibrations. 3. These variations of current strength
produce corresponding variations in the number of lines of force
passing through the secondary of the induction coil. 4. The
result is that induced electrical surges are sent chasing one another
along the line ^ire to the distant receiving instrument. 5. These
electrical waves, surging backwards and forwards in the little
coils of the receiver, produce corresponding variations in their mag-
netic field. These variations in the number and direction of
the lines of force that pass through the coils cause mechanical
vibrations of the diaphragm D', 6. Finally, the vibrations of the
diaphragm are transmitted to the air, reproducing all the modula-
tions of tone quality, pitch, and loudness that belong to the
sounds emitted by the speaker.

SUMMARY

1. The electrical energy of a battery is obtained by consuming
zinc, and the voltaic cell is therefore too expensive for generat-
ing electricity on a large scale.

2. Many physical processes are reversible, and we find that

INDUCED CURRENTS 261

this is true of the conversion of electrical energy into mechanical
work.

3. Whenever the number of lines of force that pass through a
conducting circuit is changed, an induced current is started,
which lasts only while the change is going on. The magnitude
of the induced electromotive force is proportional to the rate of
change in this number of lines of force, and its direction is always
such that its reaction — either mechanical or electrical — opposes
the change that causes it.

4. The dynamo electric machine is a contrivance for the
conversion of mechanical energy into electrical energy. It con-

.sists of a field magnet and an armature, one or the other of which
rotates, and a sliding contact device.

5. In the direct current dynamo the sliding contact device is
a commutator and brushes; in the alternator it is a pair of col-
lecting rings and brushes.

6. Each coil of the annature, as it rotates, generates alter-
nating induced currents, which are sent out through collecting
rings, or a pair of commutator segments, and thence by a pair
of brushes to the external circuit wherein the electrical work is to
be done.

7. The field magnets of a direct current dynamo are excited
by the current from its own armature, and they may be series
wound, shunt wound, or compound wound.

8. The field magnets of an alternator must be excited by a
separate direct current dynamo.

9. An induction coil consists of a primary coil which contains
a soft iron core and is surrounded by a secondary coil. The
primary current is made and broken by a contact breaker. Such
a coil may be made to give long, powerful sparks.

10. Alternating current "step-up" transformers are used to
transform currents of low pressure and great quantity into cur-
rents of higher pressure and smaller quantity for transmission
to a distance. "Step-down** transformers are used to reconvert
these into currents of low pressure and large quantity for use in
lamps and motors.

11. A telephone transmitter consists essentially of a diaphragm,

262 PHYSICS

a microphone contact, and a small induction coil, in circuit with
a local battery or other source of direct current.

12. A telephone receiver consists of a steel magnet, a small
coil (or two coils) of wire, and a diaphragm.

13. Sound impulses are transformed by the transmitter into
electrical waves, which are sent along a wire to a distant receiver.

14. The receiver reconverts these electrical waves into soiind
impulses similar in every detail to that which was spoken against
the diaphragm of the transmitter.

QUESTIONS

1. Mention some physical processes that are reversible.

2. Describe the four ways in which currents may be induced in a
coil of wire by means oT a steel magnet.

3. In each of these cases, what is the kind of f>ole induced at the
end next the magnet, the direction of the force between the induced
and hiducing poles (i.e., attraction or repulsion), and finally, the effect
on the motion (i.e., assistance or opposition)?

4. In a similar manner, describe the eight different ways in which
an induced current can be started in a secondary coil by means of a
primary coil and a battery.

5. What variation in these effects will result from the use of a soft
iron core? From increasing the suddenness of the motions?

6. Explain why the electromotive forces of induced currents are of
short duration.

7. State the lav/r, of induced currents, in which the results of all such
experiments arc summed up.

8. What are the essential parts of a dynamo-electric machine?
Point out the resemblance in construction, and the difference in action,
between a direct current dynamo and a direct current motor.

9. Briefly explain how the act of rotating an armature coil sets up
therein an induced current which changes direction at every half turn.

10. Explain ho'^V these alternating currents may be led out to the
external circuit as alternating currents by means of collecting rings,
or converted into direct currents by means of a commutator.

11. How are the magnitude and the uniformity of such direct
currents affected by having the armature made up of several coils
instead of one?

12. Mention advantages gained in the design of a dynamo by the
following features: (a) increasing the number of field poles; (6) per-
fectly balancing the coils electrically and mechanically; (c) slotted
armature core; (d) perfect insulation; (e) good ventilation.

INDUCED CURRENTS 263

13. Describe the three modes of field magnet windings.

14. Explain how a self-excited machine builds up its own mag-
netic field.

15. What are magnetos, and what are some of their uses?

16. Diagram an induction coil, trace the primary current through
its circuit, and show how the induced currents of the secondary coil
are started. Explain the action of an automatic break hammer.

17. Describe the construction and operation of the alternating
current transformer. State the relation of the electrical pressures to
the numbers of turns in the two coils. What is the great advantage of
such transformers in alternating current lighting and power circuits?

18. Diagram a telephone circuit with a transmitter and receiver
at each end of the line.

19. Trace the local current in the transmitter, and describe the
manner in which its strength is varied in correspondence with the
variations of air pressure due to the sound. Describe the induced
currents that result from these variations of the local or primary current.

20. Describe the results of these induced currents when they reach
the receiving telephone.

PROBLEMS

1. Examine a "spark coil," or "kicking coil," such as is in common
use for lighting gas by electricity. It has a soft iron core and only one
coil, of many turns. When placed in a circuit with several battery
cells, it gives a strong spark on breaking the circuit, whereas if the wire
were not coiled no spark could be obtained. Does this imply' that an
induced current is generated "at break" which adds its electrical
pressure to that of the battery? May a current in each turn of this coil
induce a current in every other turn? On "making" the circuit, would
the induced current be in the same direction as the battery current, or
in the opposite direction? This added current is called a self-induced
current. Does it differ essentially from any other induced current?

2. A break hammer induction coil is subject to troublesome spark-
ing at the break. Is this spark due to the same cause as that of the
kicking coil? This spark, by forming an arc, like that of an arc light,
bridges the gap, and not only burns the contacts, but also prolongs the
time of breaking the primary circuit. Will the induced current in
the secondary be as strong as if the arc were not formed? The arc may
be partially prevented by connecting the opposite coatings of a con-
denser across the gap. Why?

3. When two telephone wires, whose circuits are completed through
the ground instead of by return wires, are placed on poles parallel to
one another, the conversation on one line may be heard on the other.
Can you explain why? The Jipise of the trolley cars and of telegraph

264 PHYSICS

instruments is often heard in the telephone. May this be due to the
same cause?

4. Look carefully at Plate III. Do you see a dynamo direct-
connected to it? Find the iron-clad field magnet, and the armature
Also examine Plate VII. This is an alternating current generator.
Find the armature and the field poles. In this dynamo, is it the field
or the armature that revolves?

SUGGESTIONS TO STUDENTS

1. If you have made yourself a galvanometer, as was suggested in
Chapter XI, repeat the experiment of Faraday's ring (Art. 243); and
also one made by Henry at about the same time (c/. Hopkins's Ex-
perimental Science, pp. 467-476).

2. Visit the power house of the electric lighting company or the
street railway company, and make a brief report on the results of your
investigation. (In most cases it will be necessary to write a letter to
the manager, stating why you wish admittance and requesting the
favor of a pass.)

3. Belt a toy motor to the flywheel of a sewing machine, so that
you can rapidly turn the armature by means of the treadle. Connect
the terminals of the motor with your galvanometer and work the
treadle. Does the galvanometer indicate that the motor is operating
as a dynamo? If you have not made a motor you can buy one at an
electrical supply store for a dollar or less.

4. Unscrew the diaphragm end of a receiving telephone case, re-
move the diaphragm and look at the end of the magnet and its coil.

5. Read Tyndall's Faraday as a Discoverer (Appleton, N. Y.), and
S. P. Thompson's Life of Faraday (Macmillan, N. Y.).

6. Look up the biography of Joseph Henry. See, "A Study of the
Work of Faraday and Henry,*' by Mary A. Henry {Electrical Engineer,
N. Y., Vol. 13, p. 28), also Cajori's History of Physics. . This last book
will give you references to books on the lives of all the discoverers in
Physics.

7. Forbes 's Elementary Lectures on Electricity and Magnetism (Long-
mans, N. Y.), and Wright's The Induction Coil in Practical Work (Mac-
millan, N. Y.), are especially interesting in their descriptions of the
phenomena of induced currents.

8. If Faraday's Experimental Researches in Electricity is in your
city library, read some of it. It is one of the most remarkable books
that was ever written. It will be worth your while to become at
least a little acquainted with the mind of this great man.

CHAPTER XIII

THE ELECTRIC CURRENT AT WORK

248. ftuestions for Further Study. In the preceding chap-
ters, we learned how a dynamo and a motor work; but some ques-
tions of great interest still remain unanswered. How do arc
lamps and glow lamps work? How much power is required to
operate them? What are the methods of distributing the cur-
rent? How do electricians measure currents and calculate their
power? How much loss is caused by the resistance of the wires?
What uses can be made of the heat developed by the current?
How is electroplating done? How does a storage battery differ
from a voltaic battery? These are things that everybody wants
to know something about, and a little further study will enable
us to understand them.

249. The Arc Light. In 1808, Sir Humphry Davy, the
predecessor of Faraday at the Royal Institution, produced the
first arc light with a powerful battery and two pencils of carbon.
When these two carbons, connected with the terminals of the
battery, were brought into contact and then slightly separated,
the current was not broken, because an arc, composed of white
hot vapor of carbon, was formed. The carbons, being in contact
with the air, are gradually burned up, just as is the carbon in
burning illuminating gas or oil. In all arc lamps an ingenious
arrangement of electromagnets "feeds" the carbons together auto-
matically as fast as they bum away (c/. Art. 270).

250. Pressure and Current in the Arc Lamp. An ordinary
street lamp is equivalent nominally to 2000 candles. It is found
not to bum satisfactorily unless it is fed by a current having a
constant strength of about 9.5 amperes, the electrical pressure

265

266 PHYSICS

at its terminals being maintained at about 50 volts. Lamps of
greater candlepower must, of course, have more current. We
have already learned something in a general way about cur-
rent strength, resistance, and pressure; but if we wish to know
anything of the way in which electrical calculations are made,
we must learn how to express our ideas more precisely. Some
definitions and precise statements of relations, therefore, are
necessary.

251. The Current Strength in a conductor is the rate of flow
of the current, i.e., the quantiiy of electricity passing per second
at a given cross-section of the conductor; and the ampere, the
practical unit of current strength, is defined as the steady current
which deposits silver by electrolysis from a solution of a silver
salt at the rate of .001118 grams per second (c/. Art. 208).
The ampere is named in honor of Andr6 Marie Ampere (1775-
1836),. who was professor in the Polytechnic School at Paris, and
who followed up the discovery of Oersted with valuable researches
on relations between currents and magnets.

252. Eesistance. We are accustomed to conceive that a
conductor offers resistance to the passage of a current of elec-
tricity, and that an electromotive force, or electrical pres-
sure, is required to force the current through it, because in trans-
mitting the current, the conductor absorbs some of the electrical
energy and gives off this energy again in the form of heat. The
unit of resistance is the ohm, which is defined as the resistance at
0° C. of a column of pure mercury 106.3 centimeters long and
of a uniform cross-sectional area of 1 square millimeter. Such a
column should weigh 14.45 grams. The ohm is named in honor of
Georg Simon Ohm (1789-1854), an eminent German physicist, who
was teacher of mathematics and physics at the Gymnasium at
Cologne, and afterwards professor at the University of Munich.
Ohm investigated, both experimentally and mathematically, the
resistances of conductors and their relations to current strength.
By his experiments and those of others, the following relations
have been established:

THE ELECTRIC CURRENT AT WORK 267

253. The Laws of Eesistance. The resistance of a conductor
is

1. Directly proportional to its length.

2. Inversely proportional to its cross-sectional area.

3. Directly proportional to a constant whose value depends on
the material of the conductor and on the units in which its length
and cross-section are expressed. This constant is called the
RESISTIVITY of the substance, and it represents the resistance at
0° C. of a conductor of the given substance having unit length
and unit cross-section.

4. Other things being equal, the resistance of a given conductor
depends on the temperature. For most metallic conductors, the
resistance diminishes as the temperature is lowered; and it is
interesting to note that according to some recent experiments,
this diminution appears to take place at such a rate that at the
absolute zero they would have no resistance (c/. Art. 123). For
carbon, and for those substances that are broken up or electrolyzed
when conducting a current, the resistance is diminished by raising
the temperature.

264. Ohm's Law. One of the most important contributions of
Ohm to our knowledge of electric currents is the law known by
his name, and stated as follows: The current strength in any
conducting circuit is directly proportional to the electric
pressure or electromotive force, and inversely proportional to the
corresponding resistance. Letting C represent the current strength,
E the electric pressure, or electromotive force, and R the

resistance, we may express Ohm's law by the equation C = -^ or

^ .. Pressure in volts ,^^^

Current m amperes = ^ — ;— -. ^ . (11)

^ Kesistance m ohms ^ ^

This equation defines the volt for us ; for if the current strength
and resistance are each made equal to 1 unit, the pressure, ac-
cording to the equation, is 1 volt.

A VOLT, therefore, is that electrical pressure or that electro-
motive force which will maintain a current of one ampere in a
conductor whose resistance is one ohm. The volt is named in

£68

PHYSICS

honor of Volta (c/. Art. 207). A simple voltaic cell has an

electromotive force of nearly one volt. A difference in

electrical pressure is often spoken of as a difference of

POTENTIAL.

255. Ammeters and Voltmeters. It will interest us to learn
how the current strength and pressure required by a lamp or
a motor are measured. The instruments most widely used for
this purpose are made on the principle of the D'Arsonval gal-
vanometer. The coil is pivoted in jeweled bearings, and bal-
anced against a hairspring after the manner of the balance wheel

of a watch. In all voltmeters
and ammeters, the number
of volts or amperes is indi-
cated by a pointer attached
iT, to the coil and moving over
a scale which has been ac-
curately graduated or "cal-
ibrated " by reference to currents of known
strength, so as to read volts or amperes. The
voltmeter has a very high resistance, and the
ammeter a very low resistance; the voltmeter
is connected as a switch or "shunt" across the
terminals of the lamp or motor, where the
pressure is to be measured; the ammeter is
placed "in series" in any part of the circuit
around which the current is flowing. Fig. 150 shows the proper
method of connecting them. Fig. 151 shows a portion of the
switchboard in a power-house, with ammeters, voltmeters, and
switches. •

Fig. 150. Voltmeter
AND Ammeter Con-
nections

256. To Calculate the Power. The next problem that claims
our interest is the calculation of the power used in the lamp. In
Chapter IX we learned that we could calculate the amount of work
done per second in the cylinder of a steam engine, not only by
taking the product of the average force of the steam and the distance
traversed per second by the piston, but also, more conveniently,

THE ELECTRIC CURRENT AT WORK

269

by taking the product of the average steam pressure and the

quantity (volume) of steam used per second. It is easy* to show

that similarly the work done per second by an electric current is

proportional to the product of the electrical pressure and the

quantity of electricity

used per second. If we

measure the pressure in

volts and the quantity-

per-second, or current

strength, in amperes, it

is obvious that the power

will be one unit when the

pressure is one volt and

the current strength one

ampere; therefore, the

following definition is

adopted for the unit of

power:

The unit of electrical
power or activity is the
power of a current of one
ampere under a pressure
of one volt. This unit is
called the watt, in honor
of James Watt, the in-
ventor to whom we are most indebted for the modem steam engine.
A watt is found to be equal to 10^ ergs per second, or t^^^ horse-
power (c/. Art. 43). With these units, the equation that expresses
the power of a current is:

Power in watts = Current in amperes X Pressure in volts (12)

Or, if A represent the number of watts, C the current strength,
and E the pressure, A = CE,

With this equation we can now calculate the power consumed
in our 2000 candlepower arc lamp, for since C = 9.5 amperes
and E = 50 volts, the power .4 = 9.5 X 50 = 475 watts. Let the
student find the number of ergs per second and the horse-power
that correspond to this number of watts.

Fio. 161. Switchboard

270

PHYSICS

257. Watt Meters. When large quantities of electrical energy
are used in varying amounts, it is most convenient to measure the
total amount of it in watt-hours.

The watt-hour is the amount of energy furnished in one h6ur
at the rate of one watt; and it therefore equals 10^ — X 3600
sec = 36 X 10® ergs; 1000 watts is called a kilowatt (Y'^K. W.).
The number of watt-hours Used by a consumer is measured by
an interesting instrument called a watt meter. A common form
of watt meter is a little motor having no soft iron cores. Its
field coils have few turns, and are placed in series in the circuit
whose energy is to be measured; the armature coils have many
turns, and are connected as a shunt across the mains or feeders,
like a voltmeter. The instrument is ingeniously regulated, so
that the number of revolutions of the armature is proportional
to the energy supplied. Therefore, if a train of clock wheels is
geared to the armature, the total number of watt-hours of energy "
that have passed the meter up to any given time may be indicated
on a series of dials by index hands attached to the gear wheels,
just as the number of cubic feet is indicated on the dials of a gas
meter.

^ ♦.

■^

> "

—

.«. (

_„

'■ t

258. An Arc Light Plant. We now have at command the
knowledge that is necessary in making the calculations for a small

arc lighting plant. Suppose that
we wish to light a shop with ten
2000 C. P. (candlepower) arc
lamps, and must know the neces-
sary engine and dynamo power.
How shall we attack the problem?
In a case like this the lamps
are ordinarily placed in series as
represented in the diagram. Fig.
152; therefore the total resistance is the sum of all the resistances
in the circuit.

The loss in pressure, or fall of potential in any part of the cir-
cuit is proportional to the corresponding resistance, m accordance
with Ohm's law, i.e., E = CR. (cf. Art. 254). Therefore, the

i^^

Fig. 152. Lamps in Series

THE ELECTRIC CURRENT AT WORK 271

NUMBER OF VOLTS USED IN THE LAMPS is equal to that for one
lamp multiplied by the number of lamps. So the pressure needed
for the 10 lamps is 50 X 10 = 500 volts.

We must now find the voltage necessary to overcome the
resistance of the line. Since this voltage is not available for use
in the lamps, it is called the line loss or "drop." Let us suppose
that the total length of wire from the dynamo through all the
lamps and back again to the dynamo is 600 ft. The fire insurance
regulations require us to use at least a Number 14 wire to carry
9.5 amperes (Table I, page 298). This, by reference to Table I, is
found to have a resistance of 2.565 ohms per thousand feet. Since
the resistance is proportional to the length, that of 600 ft. is 0.600
of 2.565, or 1.5 ohms, nearly. The loss of pressure in the line i^,
therefore, 9.5 amperes X 1.5 ohms = 14.25 volts.

We have found that there are 500 volts required for the lamps,
and that the line loss or "drop" is 14.25 volts; hence the electro-
motive force demanded from the dynamo is 514.25 volts, and at
this pressure it must send out 9.5 amperes. The output of
power by the dynamo is 514.25 volts X 9.5 amperes = 4885
watts. Since more lamps may be needed, as the require-
ments of the shop increase, it is customary to provide for this
increase of power. Let us suppose, therefore, that we are
to order a dynamo capable of giving 750 volts and 9.5 amperes.
The power of this machine will. be 7125 watts or, say, 7.5 K. W.
The equivalent of 7.5 kilowatts in mechanical horse-power is
VjV = 10 H. P., nearly.

Since the efficiency of a good dynamo is about 90 per cent,
we must allow for a 10 per cent loss of energy in the dynamo itself.
There is also a further loss of about 5 per cent in transmitting the
mechanical power from the engine to the dynamo. Hence, 10
H. P. is 85 per cent of the power that must be furnished by the
engine to provide for the greatest load that it will get from the
electric plant. This extra power to be provided by the engine,
then, is V/ of 10 = 11.8 H. P. Our engine must, therefore, be
big enough to take care of a load of about 12 H. P. in addition to
the power furnished by it for running the machinery of the
shoj).

272

PHYSICS

259. Incandescent Lamps. The incandescent or glow lamp
consists of a slender thread or filament of specially pre-
pared carbon, enclosed in a glass bulb and mounted on a pair
of terminals that pass through a glass plug at the bottom of the
bulb.

The air is pumped from the bulb, which is then fastened into
a base. This base may be screwed into a socket in such a way
that when the terminals of the socket are connected with the
supply wires, the current is conducted through the filament. The
carbon has a relatively high resistance, and when a sufficient
current passes through it, the resulting heat makes it white hot
or incandescent (c/. Art. 153). It can not bum, however, because
the air has been removed from the bulb, and no oxygen is there
for it to combine with.

A glow lamp is usujdly so adjusted that it gives 16 candle-
power when a current of 0.5 amperes is passing through it. Its

resistance is 220 ohms; hence
the fall of pressure through it
is 110 volts, and it takes en-
ergy at the rate of 55 watts.
The student may easily verify
the last two statements by cal-
culation from equations (11)
and (12), Arts. 254, 256.

260. The Parallel Method
of Distribution. The diagram ,
Fig. 153, shows the method generally employed for distributing
the current to glow lamps.

The dynamo is designed and operated so as to maintain a
constant difference of potential at its terminals, whether the cur-
rent taken from it is large or small; and in good practice the wires
for the mains are so chosen that the pressure lost in traversing
them shall not be more than about 10 per cent of that furnished
by the dynamo.

Since the loss in pressure through the lamps is 110 volts, the
dynamo must at least have an electromotive force equal to VV

Fig. 153. Lamps in Parallel.

THE ELECTRIC CURRENT AT WORK 273

of 110 = 122.2 volts. For small plants the usual machine is wound
for 125 volts.

261. An Incandescent Light Plant. Let us suppose that a
certain man has on his country place a good-sized waterfall, and
wishes us to tell him whether he can use this power for lighting
his house. How shall we apply our knowledge of pKysics so as to.
solve his problem for him?

We must ascertain: (1) The greatest number of lamps that
will be in use at any one time. (2) The distance from the fall
to the center of distribution, or the point where the branches to
the various rooms are to be taken from the main wires or *' feed-
ers." (3) The height of the falls. (4) The least volume of
water per second that will be available for use. This makes it
necessary to measure the cross-section and velocity of the stream
above the falls.

Let us suppose that he wishes to light 200 lamps, 150 of which
are likely to be in use at any one time, and that the falls are lo-
cated 1500 feet from the center of distribution.

The lamps will require a pressure of 110 volts, and if a 125
volt dynamo is used, the line loss allowable will be 15 volts, or
12 per cent. Since the main current divides amongst the lamps
it must equal the sum of the currents in all the lamps. There-
fore the number of amperes required for 150 lamps of 16 C. P.
each, is 0.5 X 150 = 75. The output, therefore, must be 125
volts X 75 amperes = 9375 watts. In designing a lighting plant it
is always well to allow for a few more lamps, so we will figure
on a 10 K. W. dynamo, giving an output of 10,000 watts, or, in
mechanical units, -^H^ = ^^-^ horse-power.

Allowing an efficiency of 90 per cent for the dynamo, the
mechanical H. P. supplied to it by the water wheel must be:

W of 13.4 = 14.89 H. P., or, say, 15 H. P.

The efficiency of a good turbine water-wheel is about 80 per
cent, and if we allow for an additional loss of about 5 per cent in
transmitting the power from the turbine to the dynamo, the power
furnished by the water must be VV" of 15, or 20 H. P.

274 PHYSICS

Now let us suppose that the falls are 300 cm high and that
by floating ^ stick on the water just above the falls and timing
it with a watch, we find that it passes over a measured distance
of 200 cm in 2 sec. .

The velocity is, therefore, 100 ^. If in measuring the depth
and width of the stream at the place where it goes over the falls,
■ we find that a fair average of each dimension gives us, depth 60
cm, width 300 cm, the average cross-sectional area of the stream is :
18,000 cm^, and at the speed of 100 ^ the volume passing the
falls in 1 sec is 18 X 10^ cm^. The mass of the water
(cf. Art. 32) is 18 X 10^ gm, .and its weight is therefore
18 X 10^ X 980 = 1764 X 10* dynes. Since the distance
through which this weight can act is 300 cm, the potential
energy of the fall is 1764 X 10* X 300 = 5292 X 10* ergs each
second. Therefore, since 1 H. P. = 746 X 10^ ^, its horse-
power is: ^f/^^l? = 72 H. P., nearly.

We see that the inaccuracy in our method of measuring the
falls is not serious, for the result of the calculation shows us that
we have a large margin. We may, therefore, safely assume that
the stream will furnish us the necessary 20 H. P.

The sizes of wire for the feeders and branches still remain to
be calculated. We have seen that our allowable line loss is 15
volts. We must not have more than 3 per cent drop in voltage
in our distributing wires, because the drop varies with the current
used, so that when only a few lamps are lighted, it will be less than
when all are lighted. Unless the drop allowed for were small,
there would then be too much pressure for these lamps and their
life would thus be shortened. We shall therefore take 3 volts
for the drop in the distributing wires and leave 12 volts for drop
in the feeders. Now, since the line drop is to be 12 volts and
the current required is 75 amperes, we have from Ohm's law:

R = jz = -^ : — = 0.16 ohms. Since the feeders are 1500

C 75 amperes

feet long, the length of wire in them is 1500 X 2 = 3000 feet; and

the number of ohms per thousand feet is ^ J ^ = 0.0533. Consulting

the wiring tables, we find that a Number 0000, Brown and

THE ELECTRIC CURRENT AT WORK 275

Sharpens gauge wire has a resistance per 1000 ft. of 0.04966 ohms,
which is the nearest to 0.0533 ohms and is, therefore, the size to
be chosen.

The sizes of wire on each of the various branches are chosen
by a similar calculation, in accordance with the number of am-
peres taken and the length of the branch wires, so as to give a
drop in each group of nearly 3 volts, as demanded by the con-
ditions stated.

Such a lighting plant as this would be rather expensive; but
the energy would cost nothing; the repair bills would not be large,
no high-priced attendance would be necessary, and the only im-
portant cost would be the interest on the money invested.
The power could be utilized in the daytime, by means of motors,
for operating a threshing machine, feed chopper, cream sepa-
rator and churns, elevators, sewing machines, electric fans, and, in
fact, for everything in which power is needed on the place, in-
cluding the charging of electric automobiles. A few storage
battery cells could also be kept charged by the current and used
for operating door bells, burglar alarms, signal bells, and other
household apparatus, and an electromagnetic device for stopping
and starting the water-wheel by means of a switch, located in
the house.

Heating coils also might be used in the house for cooking,
ironing and the like. Thus, considering the great convenience,
cleanliness, and wide range of usefulness afforded by such an
electric plant, the capital invested in it would certainly be very
advantageously employed.

Heating Effects of the Current. In the transmission
of electric power it is desirable to have as little of the energy
transformed into heat as is possible. On the other hand, when
we want to use the energy as heat we should plan our apparatus
so as to have the electrical energy liberated in the particular
limited space where it is wanted.

It is thus very important to know definitely the relations that
the number of heat units bear to the numbers of volts, amperes,
and watts.

276

PHYSICS

These relations were investigated by James Prescott Joule, who
made the first determination of the mechanical equivalent of heat.
Joule placed a small coil of platinum wire in a calorimeter
with a weighed quantity of water and a thermometer (Fig. 154),
and in the usual manner (cf. Art. 126) meas-
ured the quantity of heat given up to the
water when currents of different strengths
were passed through the coil.

Besides measuring the pressure and cur-
rent strength, he also measured the time
intervals during which the current had been
passing. As a result of these experiments,
which were subsequently repeated with greater
accuracy by the late Professor Rowland of
Johns Hopkins University, the following rela-
tions were established :

Fig. 154. Joule's
Calorimeter

263. Joule's Law. The number of heat
units generated by a current of electricity is
directly proportional: v

1. To the square of the current strength,

2. To the resistxince of tJie conductor.

3. To the time during which the current passes.

If H represents the number of calories liberated in / seconds,

C the current strength, and R the resistance, Joule's law may be

TT
expressed by the equation, — = .24 C^R. Or,

calories in one second = .24 X (amperes)^ X ohms. (13)

The factor .24, therefore, represents the number of calories

per sec. corresponding to one watt.

p
This is apparent from Ohm's kw: for C = -^ or E = CR;

Also, from the equation for electrical power, A = CE;
so, substituting CR for E, this equation becomes A = G^R; or
watts = (amperes)^ X ohms. This, when multiplied by .24,
gives the number of calories in one second as stated by Joule's
law (c/. equations 11 and 12, Arts. 254 and 256).

THE ELECTRIC CURRENT AT WORK 277

An inspection of these equations, E = CR and A = C^R

H
(or — = .24 C^R)y will tell us what to do when we want to transmit

electrical energy with the least possible loss by heating, and also,
on the other hand, what to do when our purpose is to get heat
from the current.

264. The Heat Loss in Transmission. When considering the
energy lost in transmission, we note from Ohm's law, E = CR,
that the drop in voltage, caused by the constant resistance of the
line, is proportional to the current C. Also from Joule's law, the
energy dissipated in the line as heat is proportional to C^, There-
fore we may make these losses small by making the current as
small as possible.

Now, the power delivered by the line is measured by the
product of the current C and the pressure E at which it is deliv-
ered; and since the electrical power is the same so long as this
product CE remains constant, we can deliver the same amount
of it with less line loss by making C smaller and E larger in the
same proportion. For example, suppose we wish to deliver 1000
watts to a certain house. We can do this by means of a current
of 10 amperes at a pressure of 100 volts, or by a current of 5 am-
peres at a pressure of 200 volts. If the resistance of the line is 2
ohms, then, in the first case, the power lost in the line by heating
is, watts = C^R = 10^ X 2 = 200. In the second case the loss is
watts = 5^ X 2 = 50. Thus, by doubling the voltage, we have
reduced the watts lost in the line to one-quarter of its former
value ; and we can understand why it is more economical to trans-
mit electrical energy at a high voltage and a low amperage.

If it is desirable to save copper instead of energy, we see that
we may in the second case get the same efficiency as in the first,
using a wire of one-fourth the cross-sectional area. For if the
wire is reduced to one-fourth its former size, its resistance will be
4 times as great, or 8 ohms. Then the heat loss in the wire is,
5* X 8 = 200 watts, as at first.

From these examples we see that by doubling the voltage
while the total output of the dynamo remains the same, we can

278

PHYSICS

save either three-fourths of the energy that would be lost in
transmission, or three-fourths of the copper, as we may elect.
Thus again it appears that it is mare economical to transmit
electrical energy at high pressure and small current strength.

This superior efficiency of high voltage transmission is taken
advantage of in lighting and power circuits- in some very inter-
esting ways, two of which we will now consider.

®\

Fig. 155. The Three-Wire System

265. The Three-Wire System. Fig. 155 shows a method of
wiring much used for motors and incandescent lamps. It will

be seen from the
diagram, that if
equal numbers
of lamps are
flowing on the
two branches, the
current will flow
through the pairs
of lamps in series,
and that a given
number of lamps
will take twice the voltage, but only half the current taken by the
same number of lamps on the two-wire plan.

As shown in the preceding section, the line loss then will be
only one-fourth what it would be on the two-wire plan. If the
two sides are balanced, i.e., if they are equally loaded, no current
will traverse the middle or neutral wire. The lamps, however,
are independent of one another and of the motors. This is clearly
shown by the diagram, for if a lamp b is cut out of the negative
side, then the current from a can no longer pass through b to the
negative wire, so it returns by the neutral wire. And if one of the
lamps a is cut out of the positive side, the current necessary to sup-
ply its counterpart b, although now no longer fed to b through
a, is supplied via the neutral wire from the dynamo 7)— on the
negative side. When the two sides are balanced, the two dyna-
mos work strictly in series, giving double the voltage of one;
and the second, 2)—, sends all its current through the first, 2)-h;

THE ELECTRIC CURRENT AT WORK

279

but when the demand is greater on the negative side than on the
positive, the dynamo D — sends a Sufficient part of its current out
along the neutral wire,
and so supplies this
extra demand. With
this arrangement, for a
given number of watts
delivered and a given
line loss, the + and —
feeders can be reduced
to one-fourth the weight
required by the two-
wire plan; and since
the neutral wire has
the same cross-section
as one of the feeders but only half the length (and weight) of
the two feeders, the total weight of copper used is i + i = i of
that required by the two-wire i)lan.

Fig. 156 shows how electrical power is utilized when trans-
mitted to a large shop. The motor is direct-connected to a
lathe which is turning axles for car wheels. Figs. 157 and 158

Fio. 156. Motor-Driven Lathe

Fig. 157. Motor-Driven Shop

Fig. 158. Belt- Driven Shop '

show the advantage of transmission by wires over transmission by
belts and shafting.

280 PHYSICS

266. Alternating Current Transmission. We are now pre-
pared to appreciate the value of the alternating current trans-
former (Art. 245), for by means of a transformer a current can,
with very little loss of energy, be "stepped up" from 125, 250,
or 500 volts in the generator to 5,000, or 10,000, or even to 50,000
volts in the feeders, and then ''stepped down" again to 220 or 110
volts by the means of another transformer. Thus the current
is at a high voltage during transmission, and the losses in the
line are reduced to a minimum. These high voltage currents
are very dangerous and easily escape by leakage unless the con-
ductors by which they are carried are thoroughly insulated.

Since alternating currents can be transmitted with so much
greater economy than can direct currents, the former are rapidly
displacing the latter for most purposes.

267. Electrical Heating. Having found in the preceding
paragraphs that the heating effect of the current increases as the
resistance and as the square of the current strength, it Is clear
that when we want to convert the current energy into heat we
must have either a large current, or a large resistance, or both.
By thus converting electrical energy into heat, very high tem-
peratures may be obtained; so that the process is much used
in electric welding, and in the reduction of ores.

In the reduction of aluminum, for example, a current of 2,500
amperes, under a pressure of only about 8 or 9 volts, is passed
from a large carbon terminal, through the ore in a airbon-lined
crucible, which forms the other terminal. The reduction of the
metal from the molten ore is effected partly by the intense heat
and partly by electrolysis. Carborundum, which is much used
instead of emery for grinding edge tools, and calcium carbide,
which is used for producing acetylene gas for lighting purposes, is
made in electric furnaces.

For heating suburban cars, and also soldering tools, cooking
utensils, flatirons, chafing dishes, and even curling irons, current
at ordinary pressure is passed through coils of highly resisting
metal, such as iron, German silver, or platinoid. Electric heat-
ing, as compared with direct heating by burning the fuel without

!^ .E 3

Fig. 159. A Divided Circuit

THE ELECTRIC CURRENT AT WORK £81

transformation through a steam engine and dynamo, is altogether
too expensive to come into general use in cases of this sort,
but will always be appreciated when small quantities of heat
are needed, when the greater convenience and cleanliness offset
the dispropoiiionate cost.

268. Divided Circuits. In considering the distribution
of current in parallel conductors, in connection with glow lamps,
it is found that if the branches

have equal resistances they C | ^

get equal portions of the cur- . y^ 3 S^^^ _^

rent. In electrical engineer-
ing, it is often necessary to
know the amount of cur-
rent on each of several
branches whose resistances
are not equal; or sometimes
it is desirable to arrange the resistance of a branch or "shunt" so
as to switch off a definite fraction of the current.

We must, therefore, know how the resistances of the branches
govern the distribution of the current among them. In order to
find this relation, let us consider the conductor. Fig. 159.

It is evident from this diagram that there is a definite differ-
ence of potential, or loss of pressure, between A and B, and that
this must be the same for each branch.

Let us call this difference of potential E volts.

E volts
The current on AC.B is then C, = ^ , , or E = C, X 2,
^ ^ 2 ohms

E volts
6 ohms'
the difference of potential is the same for both branches,

c,x2 = c,x ^ = ^«r§^ = | = r

Of the 8 amperes, therefore, the 2 ohm branch gets 6 am-
peres or f , and the 6 ohm branch will get 2 amperes or J, i.e., the
current strength on each branch is inversehj propoi-tional to the
resistance of that branch. It may be proved, both mathematically

and the current on AC2B is Cj = ^ 1 ,„ > or £ = C2 X 6. Since

282

PHYSICS

and experimentally, that this is true for any number of branches
whatever.

Shunts. We may apply this principle when we are
using a delicate galvanometer with a strong current, and wish to
send only xu^inr of the current through it, so as not to injure it.
We then connect a shunt across its terminals, and the resistance
of the shunt must be j^-^ of the resistance of the galvanometer
branch. The current will divide between the galvanometer coil
and the shunt as it does between AC^B and AC^B, Fig. 159;
so the galvanometer will then get j^Vtt of the current and the
shunt the remainder, or iVoV-

Tie

270. Arc Lamp Eegnlation. Another very important appli-
cation of the shunt principle is found in the regulating magnets
of the arc lamp.

The diagram. Fig. 160, shows on the reg-
ulating magnet two windings, which are car-
ried around it in opposite directions, one of a
few turns of thick wire and in series with
the carbons, and the other a shunt coil of
many turns of fine wire. When the current is
turned into the lamp, the carbons are in
contact, and their resistance is small; so that
the current is cori-espondingly large. This cur-
rent, going around the series coil, magnetizes
it strongly. This causes it to pull up the core
C, which in its turn acts on the clutch K so
as to pull up the + carbon. When the +
carbon is pulled up too far, the current
through the carbons is reduced, because of the
greater resistance of the lengthened arc; but
because of this increased resistance a greater
proportion of the supply current'goes around the shunt coil. This,
being wound in the opposite direction, counteracts the magnetic
pull of the series coil and lets the clutch down, so that the car-
bon slips through it and the arc is shortened. If the arc gets

C?*

^^^

FiQ. 160. The Arc
Lamp Regulator

THE ELECTRIC CURRENT AT WORK

283

too short, the diminished resistance allows more of the current to
go through the series coil* and less through the shunt coil, so

the series coil pulls the clutch
up and lengthens the arc.

If the resistances of the
two coils are properly pro-
portioned, the carbons will
be kept a constant distance
apart, as mentioned in Art.
240. This principle of di-
vided circuits has important
applications in the shunt
windings of the field mag-
nets for dynamos and motors, and in connection with controllers
used in starting motors.

Fig. 161. Section of a Lifting Magnet

271. Lifting Magnets. Another important application of elec-
tro magnets remains to be considered. This is the lifting magnet.
Fig. 161 represents a cross-section
of one, showing how the poles are
arranged so as to have a conaplete
magnetic circuit. Anyone who has
seen one of these magnets in opera-
tion, lifting heavy steel plates or gird-
ers and carrying them about a shop,
will appreciate their convenience.
Fig. 162 is a photograph of such a
magnet holding a mass of iron that
weighs about 5 tons. This mass of
iron is called a "skull cracker,*' and
it is used to break up old castings
before remelting them. The mass is
lifted by means of the magnet and

then dropped on the castings. Fig. 162. The Lifting Magnet

AT Work

272. Voltaic Cells. We have found the answers to most of
the questions that were raised at the beginning of this chapter,

c z

FiQ. 163. Batteries i^ Series

284 PHYSICS

and we shall now try to answer the others. Of the discovery of
the voltaic cell and the important fact that it supplies a continuous
current of electricity, we learned something in Chapter XI. But
we were then interested chiefly in the discoveries in electromag-
netism, which immediately followed the discovery of currents,
and so we deferred the study of voltaic and electrolytic action
until now.

Battery cells, as well as other electric generators, must have a
COMPLETE CONDUCTING CIRCUIT in order to do work. Fig. 163

shows a good way of repre-
senting a circuit in which, for
example, an electromagnet
is operated by several cells
in series (c/. Art. 258 and
Fig. 152). In such a case it
is found that both the volt-
ages and the resistances of
the circuit are added up, which is what we should expect, since the
current goes through one cell after another. It follows, therefore,
that if we have any number of equal cells in series, with some
external resistance:

(1) The total electrcmiotive force of the circuit is that of a single
cell multiplied by the number of cells.

(2) The resistance of the battery itself, which is called the in-
ternal resistance, is that of a single cell multiplied by the number
of cells.

(3) The total resistance of the circuit is the internal resistance
plu^s the sum of all the external resistances.

(4) The current strength, therefore, may he found by dividing
the total number of volts E. M. F. by the total number of ohms
resistance,

273. Energy of the Cell. When a voltaic cell is sending a
current around a circuit, it will be noticed that chemical changes
are going on. Bubbles of hydrogen gas are seen to be Hberated
at the copper plate. Also, if we weigh the two plates before
operating the circuit, and after some time remove them and

THE ELECTRIC CURRENT AT WORK 285

weigh them again, we shall find that the zinc plate has diminished
in mass, while the copper plate has not.

If we were to determine the chemical composition of the fluid,
we should find less sulphuric acid in it than before, and we should
find a new compound, zinc sulphate. The relations between the
quantities of the substances taking part in this chemical change
are found to be constant. Thus, for every 65 grams of zinc that
disappear, 98 grams of sulphuric acid are used up. At the same
time 2 grams of hydrogen are liberated, and 161 grams of zinc
sulphate are made. These relations may be represented by
symbols, as follows :

and produce and

Zinc Sulphuric acid Zinc Sulphate Hydrogen

Zn +H2SO, =ZnSO, H^

65 gm 2+32+64=98 gm 65+32+64=161 gm 2 gm

The s3mibols, S and O^ represent sulphur and oxygen in the
proportions of 32 and 64 gms, respectively, to 2 of hydrogen or
65 of zinc. From an inspection of the relative weights of the
substances, it is evident that the zinc replaces the hydrogen in
the sulphuric acid and combines with the SO4 (or sulphion, as
it may be called).

Now, we know that we get energy from carbon by burning it,
i.e., causing it to combine with oxygen from the air. Similarly,
it ought now to be quite plain that we get energy from the voltaic
cell by causing zinc to combine with the oxygen in the sulphine of
sulphuric acid, as was stated in Art. 229.

Thus chemical energy is transformed into electrical energy.

274. Polarization of a Voltaic Cell. A very troublesome
defect in Volta's cell is that the hydrogen bubbles stick to the
copper plate and diminish the current in two ways:

1. Hydrogen is a bad conductor, as all gases are, and it insu-
lates that part of the plate which it covers, so the internal resist-
ance is increased.

2. It tends to recombine with the acid and displace the zinc.
This tendency gives rise to an electromotive force which tries to
send a current backwards through the cell. This counter electro-

286

PHYSICS

motive force weakens the current,
cell is said to be polarized.

When in this condition, the

275. Why is the Hydrogen Liberated at the Copper Plate ?
We have seen that the metal which combines with the SO4 and
releases the hydrogen is the zinc, not the copper; for the zinc is
found to be used up, while the copper is not. It seems, then,
that although the chemical action starts at the zinc plate, yet the
copper plate is the one at which the hydrogen is actually liberated.
In order to explain this, a hypothesis has been advanced, which
will be understood by reference to Fig. 164.

It is supposed that when the sulphuric acid is dissolved in the
water, some of its molecules are always broken up into parts,

each of which carries a
charge of electricity, one
kind of part being positively
charged hydrogen; the other
kind, negatively charged sul-
phion.

It is conceived that the
solution is in an unstable
condition. Some of the
molecules arc always broken up into these electrically charged
parts, or ions, as they are called, and the ions are contin-
ually recombining to form molecules. If this is really the case,
it is easy to see that as a result of electrical attractions between
the oppositely charged ions, a series of combinations would take
place, as indicated by the arrows in the diagram, and there would
be a procession of positively charged hydrogen ions toward the
copper, and of negatively charged sulphions toward the zinc.

276. The Ion Hypothesis. This hypothesis not only gives us an
idea as to why the hydrogen comes off at the zinc, but also gives us
some conception of a mechanism by which electricity may be con-
ducted through a liquid and how the liquid may be broken up while
conducting the current. It is also found to fit well with the molec-
ular hypothesis, which we found it convenient to adopt while

Fio. 164. The Ions Change Partners

THE ELECTRIC CURRENT AT WORK

287

studying heat, for it explains many phenomena of solutions,
which would be very difficult to understand otherwise.

The plate toward which the + ions travel, the copper in this
case, is called the cathode, and the one toward which the —
ions travel, the zinc in this case, is called the anode. The direc-
tion of the current is conceived to he from anode to cathode through
the liquidy and from cathode to anode along the imre.

277. Local Action. Another defect of the simple voltaic
cell arises from the fact that the zinc continues to waste away,
even when the circuit is open, so that no current is passing.
This waste may be almost entirely prevented

by amalgamating the zinc with mercury.

The reason for this will be understood from
Fig. 165. Millions of little particles of carbon
and iron exist as impurities in the plate,
and, with the neighboring zinc particles, they
form diminutive voltaic cells which give risu
to little local currents, as represented by the
arrows in the diagram. As these little cur-
rents travel in short circuits, and never get out
to the conducting wires, all their energy is
transformed into heat and is wasted.

The mercury prevents this local action
by dissolving zinc to form a zinc amalgam, which Spreads over
the zinc plate and covers up the impurities.

278. Commercial Cells. Very early in the history of the
simple voltaic cell, it was found to be inefficient. Accordingly,
several modifications have been made to increase the output.

1. Since the voltage depends on what substances are used tor
the plates and the active fluid, various metals and fluids have
been tried. Carbon or copper is now used almost exclusively
for the anode, zinc for the cathode, and either sulphuric acid
or ammonium chloride (sal ammoniac) for the active fluid.

2. The internal resistance is diminished (a) by giving the
plates as large a surface as ])ossible, and (b) by diminishing the
distance between them? Why?

(

Fig. 165. Local
Action

288

PHYSICS

3. Polarization is remedied by using some oxidizing agent,
i.e., a substance that easily breaks down chemically and gives
off oxygen, which combines with the hydrogen to form molecules
of water. Polarization is prevented by introducing a second
fluid, which deposits a metallic ion instead of hydrogen.

279. The Chromic Acid Cell. This is the best kind of cell
for amateur use and for schools that are not equipped with a
direct current dynamo, or storage battery. The anode is carbon,
the cathode zinc, the active fluid sulphuric acid, and the depolar-
izing agent chromic acid.

280. The Leclanche Cell. Many forms of this cell are sold
under various trade names. The anode is carbon, the cathode
zinc, the active fluid ammonium chloride, and the depolarizing

substance a black powder, called manga-
nese dioxide. This is compacted around
the carbon, or inclosed with it in a porous
cup of either carbon or earthenware. This
type of cell polarizes very rapidly in spite
of the action of the manganese dioxide,
and hence should never be used in a cir-
cuit that is to be kept closed for any consid-
erable time. It is a very good cell for
door bells, electric gas lighting, local bat-
tery for telephone transmitters, etc., when
the current is used during short aiid in-
frequent intervals. The so-called dry cells
are of this type.

Fio. 166. Gravity Cell

281. The Gravity Cell. This is a two-
fluid cell. The anode is zinc, and is placed
at the top of the cell; the cathode is copper,
and is placed at the bottom. The active fluid is very dilute sul-
phuric acid, and floats on the depolarizing fluid (a solution of
copper sulphate), which is heavier than the sulphuric acid, and so
remains at the bottom. Polarization can not occur when the cell

THE ELECTRIC CURRENT AT WORK 289

is in good condition, as will be seen by reference to Fig. 166. This
diagram shows that since the copper cathode is surrounded by
a copper sulphate solution, there are no hydrogen ions to be
liberated there. Copper ions are liberated instead, and these
can do no harm.

Gravity cells must be kept in a closed circuit with a large
external resistance, and if the zincs are large and an excess of
crystallized copper sulphate is kept in them, they last for a long
time and without attention, and give a very steady current.
They are in common use for telegraphic work, except on long
distance lines, where dynamos are now used.

282. Electrolysis. In the voltaic cell, ions combine and give
up their charges, producing an electric current. This process is a
reversible one, for if a current from a dynamo or voltaic battery
be sent between two plates of metal immersed in a solution that
contains a salt of any metal, the ions of this salt immediately begin
to progress in opposite directions, the metallic or positively charged
ions going toward the cathode, and the non-metallic, or negatively
charged ions, going to the anode.

This process of breaking up a liquid compound by passing a
current through it, is called electrolysis (c/. Art. 208).

The liquid is called an electrolyte, or electrolytic con-
ductor. The anode and cathode plates are called electrodes.
The relations which exist in connection with electrolysis were first
thoroughly investigated by Sir Humphry Davy, who discovered
metallic sodium and potassium by this process, and by means of
it, made many important advances in chemical science. Davy's
work was followed up and greatly extended by Faraday, who
discovered and announced the following relations:

283. Faraday's Laws of Eleotrolysis. 1. The amount of
chemical action is the same in all parts of the circuit.

2. The mass of any kind of ion liberated at an electrode is pro-
portional to the quantity of electricity that passes, i.e., to the product,
amperes X seconds.

3. The mass of any kind of ion liberated by a given quantity
of electricity is proportional to its chemical equivalent, i.e., to the

290

PHYSICS

number of grams of it that combine with a gram of hydrogen (or
replace one gram of hydrogen in combination).

Knowledge of the laws of electrolysis has been employed in
many ways, some of the most interesting of which will be described
presently, but none of its applications is so important as its use
as a means of investigation in theoretical chemistry, upon which
all practical chemistry is based. Among the other uses of elec-
trolysis we may mention the measurement of current strength
for the purpose of standardizing galvanometers, voltmeters, and
ammeters, the reduction of metals from their ores, the refining
of crude copper for electric conducting wires, the making of
electrotype plates from which books are printed, electroplating,
and the storage battery.

284. Electroplating. Fig. 167 shows an electroplating bath.
It is usually a large vat, containing a solution of some compound

of the metal which is to be

deposited. Thick copper
rods, or "bus bars," are
laid along its length, and
two of these, B+ B+, are
connected with the +
feeder from a dynamo, the
other with the — feeder.
From the + bus bars
are suspended, as anodes, large plates A A of the metal to be depos-
ited, and from the — bus bar are hung, as cathodes KK, the arti-
cles to be plated. When the current is passed, the procession
of metallic ions goes toward the cathodes, giving them the desired
coating, while for every ion deposited on the cathode, an ion from
the anode goes into the electrolyte and takes its place.

The electrolyte thus remains constant in strength, while the
anodes lose as much metal as the cathodes gain.

Fig. 167. An Ei-ectroplating Bath

285. Storage Batteries. The storage battery is the converse
of the electroplating cell; the electrolyte is dilute sulphuric acid
and the electrodes are perforated lead plates, photographs of

THE ELECTRIC CURRENT AT WORK

291

which are shown in Figs. 168 and 169. The perforations
in these plates are filled with oxides of lead. When a current
from a dynamo is passed through this cell, hydrogen ions go to
the cathode and reduce the lead oxide there to metallic lead.
Simultaneously, sulphions go to the anode and there give up oxy-
gen, which combines with the lead oxide to form peroxide of
lead, i.e., an oxide with a greater proportion of oxygen in it.
When all the oxides are thus changed, the cell is said to be fully
charged; and this condition is indicated by a copious evolution of
hydrogen and oxygen gases at the electrodes. This charged cell
is in a highly polarized condition (c/. Art. 274); it has a counter

FiQ. 168. Storage Battery, Pos-
itive Plate

Fig. 169. Storage Battery, Neg-
ative Plate

E. M. F. of polarization equal to about 2 volts, which tends to
send out a reverse current.

In fact, if 55 of these cells in series are charged by a small
110 volt dynamo, and the belt by which the dynamo armature is
driven be thrown off, this reverse current will go back through
the dynamo and run it as a motor. The storage battery, theriy is
a battery in which electrical energy w transformed into chemical
potential energy and stored np for future use. The reversed pro-
cession of the ions begins whenever the electrodes are joined
through an external conductor, and the chemical energy is re-
converted into electrical energy, ready to do any kind of work.

292

PHYSICS

Fig. 170. Storage Battery Plant

Storage batteries are used for running automobiles, but their
greatest use at present, is in large power plants, where they are
employed to store energy when the demand is light and pay it out

again when the demand
is extra heavy. Fig.
170 shows such a stor-
age battery, belonging
to a large power plant.
They have not yet come
into very extensive use
for automobiles because
of their great weight
and liability to dete-
rioration when not
properly cared for.
There ought, however,
to be a great future for
them in connection with windmills, because energy could be stored
up in them when the wind was blowing strongly and taken from
them when the wind was too light to operate the mill.

286. Eetrospect. Before leaving the study of electricity and
magnetism, it may be useful to review some of the important
things we have learned about them. We have discovered that
when two bodies, made of different substances, are brought in
contact and then separated, each is electrically charged, one posi-
tively, and the other negatively. These electrically charged bodies
have been found to act on other bodies not in contact with them;
and this action takes place not only through dielectrics, but also
through a vacuum. We have seen that a magnet acts on another,
or on a magnetic substance, although air, or wood, or other sub-
stances are between them. We have learned how an electric cur-
rent is generated by chemical action in a voltaic cell, and how
such a current is equivalent to electrostatically charged particles in
rapid motion. Finally, we found that a magnetic field may be
obtained by passing a current through a wire or coil ; and con-
versely, that a current may be produced by moving a magnetic

THE ELECTRIC CURRENT AT WORK 293

field or changing its strength in the neighborhood of a closed
conducting circuit.

All of these effects, produced by charged bodies, currents, and
magnets, take place without apparent connection between the
bodies that so act. Since it is hardly conceivable that these actions
are effected without any connection whatever, we are constrained
to assume that they are produced by stresses in some medium.
If we have to suppose that a medium exists for these phenomena,
is it not simpler to conceive that this medium and the one that
transmits the heat waves are the same? Let us then adopt the
hypothesis that these electric and magnetic phenomena are mani-
festations of stresses of some kind in the ether. We shall have
occasion to recall this h3^othesis before we finish the study of
the other branches of this subject.

SUMMARY

1. Current strength is measured in amperes, electromotive
force in volts, and electrical resistance in ohms.

2. The resistivity of a substance is the resistance at 0°,C. of a
conductor of the substance having unit length and unit cross-sec-
tional area.

3. The electrical resistance of any substance may be found
from the equation:

Resistance _ Resistivity X Length
Area of Cross-section

4. Ohm's Law is expressed by the equation:

Volts

Amperes = -tt-, .

^ Ohms

5. Joule's Law. The power consumed in an electrical con-
ductor is found by the equation :

Watts = (Amperes)* X Ohms = Volts X Amperes: the heat
developed, by the equation:

Calories = Watts X .24.

6. One horse-power = 746 watts, or 746 X 10^ |^.

7. Electricity is distributed in series and in parallel circuits;
sometimes by a combination of both.

294 PHYSICS

8. Economy of electrical transmission is secured by using high
electrical pressures.

9. Electrical power is utilized by means of motors. Elec-
trical heating is done by means of coils of high resistance, or by
suitable electric furnaces in which the heating action is somewhat
similar to that of the arc light.

10. In divided circuits, the current in each branch or "shunt,''
is inversely proportional to the corresponding resistance.

11. Delicate electrical apparatus may be used with large
currents if suitable shunts are employed.

12. In a voltaic or electrolytic cell the electricity is conducted
by a progression of + ions toward the cathode and of — ions toward
the anode.

13. The polarization of a voltaic cell may be remedied by
oxidizing the hydrogen as in the chromic acid or the Leclanch^
type; or it may be prevented by employing a second solution
from which are deposited ions of the cathode metal^ as in the
gravity type.

14. ,When cells or other electric generators are joined in
series, the total, voltage is the sum of the voltages of all, and the
total resistance is the sum of all the resistances of the circuit.

15. Faraday's Laws of electrolysis state that the mass of an
ion liberated is equal to its electro-chemical equivalent multiplied
by the product of the current strength and the time.

16. The most common and important applications of elec-
trolysis are: 1. Current measurement. 2. Electroplating and
electrotyping. 3. Reduction of metals from their ores, and the
refining of crude metals. 4. Storage batteries.

PROBLEMS

1. How many amperes are there in a current which deposits
16.77 gm silver in 50 minutes? How much silver will be deposited
by a 2 ampere current in 30 minutes?

2. What electromotive force will send a current of 10 amperes
through a lamp whose resistance is 4.8 ohms? What is the resistance
of an arc lamp which takes a current of 15 amperes at a pressure of
65 volts?

3. What is the rate (watts) at which each of the lamps of problem

THE ELECTRIC CURRENT AT WORK 295

2 consumes energy? Find the mechanical equivalent (horse-power
hours) of the energy used by each in 12 hours.

4. Find the voltage and the power (K. W.) of a dynamo that
will operate a series arc lamp circuit, the data being as follows:
Number of lamps, 25; volts for -each lamp, 45; current strength, 10
amperes; length of line circuit 2500 ft.; resistance of wire (ohms per
thousand ft.)f 2.5. What horse-power must the engine supply to the
dynamo, if we allow an efficiency of 80 per cent for the dynamo and
belt?

5. How many gm cal of heat may be obtained from a current of
2000 amperes at a pressure of 12 volts? How many Kg of copper
may be raised from 14° C. to its melting point (1054°) by this heat
if the mean specific ht. of copper for this temperature range is 0.105?

6. The following problem illustrates the way in which automobile
engines are tested in a certain large factory. A car was lifted on jack
screws, and the driving wheels were belted to a dynamo. When
the angular velocity of the drivers corresponded to a linear car speed
of 25 ^^-^ the dynamo was able to light 128 16-candle-power glow
lamps connected in parallel circuit. The meters showed that the
lamps were taking half an ampere each at 110 volts pressure. They
were so near the dynamo that there was no line drop. What were
the total current strength and the electromotive force of the dynamo?
Its output (watts)? What H.P. did it take from the engine, allowing
for 25 per cent loss in the dynamo and belt? The answer corre-
sponds to the horse-power developed by the engine when the car has
the given speed on a smooth, level road.

7. A waterfall 6.10 m high delivers 15,000 Kg water per minute.
What is its H.?.? What H.P. may be delivered from it by a water-
wheel having an efficiency of 70 per cent through a dynamo having
an efficiency of 85 per cent? The output of this dynamo will equal
how many watts? If its electromotive force is 125 volts, what
current will it supply?

8. The same fall, measured in foot and pound units, gives; pounds
of water per minute 33,000, height 20 ft. Answer the questions
and compare the answers with those of question 7.

9. The dynamo, problems 7 and 8, delivers its current to a num-
ber of glow lamps in parallel, with a 10 per cent drop in voltage
through the feeders. What is the voltage through the lamps? The
current delivered by the feeders? Allowing 0.5 amperes per lamp,
how many lamps may be operated? Calculate the line resistance
and the watts lost in the line, from the data here given. If the line
is 500 feet long, consult the wiring table, p. 298, and find the gauge
number of the proper size of wire to use?

10. Suppose in the lamp circuit, problem 9, a pair of branch

296 PHYSICS

wires have a length of 100 ft. and supply 20 lamps, what current
must they carry? They are to be chosen so as to have a 2.5 volt
drop. What is their resistance? From the wiring table find the
gauge number to be selected. Find the rate (watts) at which
energy is lost in these branch wires, and in the feeders. Find the
heat developed (gm cal) in each case.

11. An electric bell has a resistance of 10 ohms and works per-
fectly when connected with short wires to one dry cell having an
electromotive force of 1.5 volts, and an internal resistance of 0.3
ohm. What current is it then using? The same bell is connected
on a door bell circuit of 150 ft. of number 20 copper bell wire, and
although the current is found to be complete, it does not work.
Mention two ways in which the trouble might easily be remedied.
From the wiring table find the resistance of the line wire and again
calculate the current strength. Suppose two more cells just like
the one in the last problem were connected in series with it and the
circuit. Calculate the resulting current.

12. In table II, p. 299, the resistivities of some metals are given,
the resistivity for this table being defined as the resistivity in ohms
of a conductor one meter long and one square millimeter (1 mm')
in cross-sectional area. From this table and the laws of resistance
(Art. 253), find the resistances of the wires for which the following data
are given: 1. A silver wire 3 m long and O.l mm in diameter. 2.
A German silver wire 20 m long and 0.5 mm' in cross-section. 3. A
lead wire 0.6 m long and 0.8 mm' in section.

13. From the table of resistivities express the resistances of each
of the materials in terms of the resistance of copper as a unit; thus,
other things being equal, the resistance of a platinum conductor is
how many times that of a copper one?

14. A galvanometer has a resistance of 5 ohms, and it is desired
to use it with a current of 10 amperes; but the greatest current that can
be sent through it without either injuring it or causing its deflections
to be too great to read is 0.1 ampere. What must be the resistance of
a shunt that will produce the desired result when connected across
its terminals?

15. In Fig. 171 a current divides at A and reunites at B. A gal-
vanometer G is connected across, as shown. If the current on ACxB
is Cj, and that on A C2 B is C72, show that the pressure on the part
A Ci is ei = CiTif and that on the part A C2 it is 63 =C2r3. Now, if
the point of contact C2 be moved along A C2 B toward A or B till a
place is found such that no deflection of the galvanometer occurs,
show that under this condition the points Ci and C2 are equi-potential;
i.e., t'lere is no difference in electrical pressure at these two points.
Prove that when thb is the case 63 = Cg, hence CiTi =5 C2rz (o). In a

THE ELECTRIC CURRENT AT WORK 297

similar way, prove that under the same conditions Cir2 = C2r4 (&).
Divide the equation (a) by the equation (6) and simplify. What
relation has been proved to exist among the four resistances, when no
current passes through the galvanometer?

16. Suppose in the last probleni the resistances r2,r3,r4 can be varied
at will, and that r^ is a certain unknown resistance whose value we
wish to find. For example, we

make ra = 1 ohm, r^ = 100 ohms,
Ti the unknown, and then begin to
vary r2 until we have found a
value for it which will leave the
four resistances so adjusted that
no current passes through the gal-
vanometer. Now Ti is the only
unknown quantity, and the equa- Fig. 171

tion obtained in the previous

•problem applies, for the conditions are the same as there supposed.
What is the numerical value of ri when r2 = 250?

17. Show that if AC B is made of a wire of uniform cross-sectional
area and material, the ratio of the lengths of the two parts A C2 and
C2B may be substituted for the ratio of the actual resistances, and
give the same numerical value as before for the unknown resist-
ance. Apparatus arranged to measure electrical resistance in the
ways suggested by problems 15-17 is called a Wheatstone bridge, and is
of great service in practical electrical work as illustrated in this chapter.

18. Does the electromagnet, Figs. 161, 162, really do any "lifting,"
or does it simply hold while the crane-hobt to which it is attached
does the lifting? Then does such a magnet use a great amount of energy?

SUGGESTIONS TO STUDENTS

1. If you are permanently interested in electricity or expect to go
farther in the study of it than this course can take you, you ought to
own these, two books; Elementary Lessons in Electricity and Magnetism,
by S. P. Thompson, and Elementary Electricity and Magnetism, by D. C.
and J. P. Jackson (both published by Macmillan, New York).

2. Find out from books, or the bulletins of electrical manufacturers,
or from some friend who is an electrician, what a "booster" or rotary
transformer is, and how it is useful in connection with a storage battery
for an electric railway power plant, or an electric light station. If you
can get the information, write a short paper about it.

3. From similar sources, get information about the use of a starting
resistance with a shunt motor, as shown at -B, Fig. 135, and put the
results into the form of a written report. Can you find out anything
about the counter-electromotive force of a motor in this connection?

298

PHYSICS

About the danger of a "field discharge" when a shunt motor is sud-
denly shut off, without having in circuit with the armature a suitable
starting resistance like that mentioned? See Practical Electricity,
by J. C. Lincoln (published by the Cleveland Armature Works, Cleve-
land, Ohio), which is excellent on the subjects treated in this chapter.

4. Find out also how the electromotive force of the shunt coils
of a shunt or compound woimd dynamo may be regulated for varying
loads by means of a variable resistance, placed in the same branch of
the circuit with the shunt coils of the field magnets, as in Fig. 150.

5. Visit an electrotype foundry or electroplating shop, find out
what you can and make a report.

6. Find out what you can about electric elevators and hoists.

TABLE I
Characteristics op Copper Wire

Number.

Diam-
eter.

Cr08.S-

Section.

Weight

Resist-
ance.

Amperes.

Diameter.

Num. •

BER.

Brown &
Sharp's
Gauge.

MUs

Imil =

.001

inch.

Circular
Mils. d2.

Pounds

per
1000 ft.

Ohms per

1000 ft.

at 75» Fahr.

= 24«C.

Safe

Carrying

Power.

Milli-
meters
1mm =
0.001 cm

Brown

&
Sharp's
Gauge.

0000

460

211600

641

0.04966

210

11.684

0000

000

410

168100

509

0.06251

177

10.405

000

365

133255

403

0.07887

150

9.266

325

105625

320

0.09948

127

8.254

289

83521

253

0.1258

107

7.348

258

66564

202

0.1579

6.544

229

52441

159

0.2004

5.827

204

41616

126

0.2525

5.189

182

33124

100

0.3172

4.621

162

26244

0.4004

4.115

128

16384

0.6413

3.264

102

10404

1.01

2.588

6561

1.601

2.053

4096

12.4

2.565

1.628

2601

7.9

4.04

r291

1600

4.8

6.567

1.024

1024

3.1

10.26

Rubber

0.812

20.1

404

1.2

26.01

Covered
Insulated

0.511

12.6

158.8

0.48

66.18

Wire. •

0.321

10.0

100.0

0.30

105.1

0.255

8.0

64.0

0.19

164.2

0.202

THE ELECTRIC CURRENT AT WORK 299

The following table gives the resistivities of some metals, the re-
sistivity for this table being defined as the resistance at 0° C of a con-
ductor of the metal having a length of one meter and a cross sectional
area of one square millimeter. The resistivity of German silver and
other alloys varies with the composition.

TABLE II
Resistivities

Ck)pper 0.017

Silver 0.016

Platinum 0.108

Lead 0.210

Iron 0.100

German silver . 34

Practical Use of Tables I and II

Electric wiremen often use a formula for determining the cross
section of a wire of given dimensions and having a required resist-
ance, and express this cross section in circular^ mils. A wire of cir-
cular cross section and one one-thousandth of an inch (1 mil) in
diameter is said to have a cross section of one circular mil. They
also define the specific resistance of a wire as the resistance of one
mil-foot (i.e. 1 mil in diameter and 1 ft. long) at 75® Fah, The re-
sistance of a mil-foot of good commercial copper wire is 10.5 ohms.
If L represents the length of a wire in feet, CM its. cross section in

circular mils, show that its resistance R=10.5 77^7 ohms. Also that
* CM

OM = — ^^— ^. Wiremen also determine the cross section of a wire
of a certain length, which wUl cause a given drop with a given cur-
rent by the equation, CM= — —y wherein CM represents cir-
cular mils, L the length in feet, A the amperes carried, and V the
drop in volts. Show from the preceding formula and from Ohm's law
that this one is correct.

From these formulas and tables I and II you can make up and
solve as many wiring problems of this sort as you like.

Another good exercise will be to find the specific resistances in
mil-foot units of the substances in table II by multiplying the re-
sistance of a mil-foot of copper (i.e. 10.5) by each of the ratios ob-
tained in problem 13, p. 296.

If you are taking the commercial course of your school, an inter-
esting exercise will be to calculate the cost of the copper, also to
get data from a friend who is an electrical contractor and draw up
specifications and estimates for the two electrical plants describ^.d in
this chapter.

. CHAPTER XIV

WAVE MOTION

287. Of Waves. In Chapter VIII, while studying the trans-
fer of heat from one body to another when there is no visible
contact between them, we were led to adopt the hypothesb that
heat energy is propagated across apparently empty space by some
form of invisible wave motion. Similar reasons lead us to adopt
a wave hypothesis to describe the phenomena of sound and light,
and we can best appreciate how beautifully this theory fits all the
facts together into an intelligible story, if we first make a little
study of the waves with which we are al! /amiliar. Let us, then,
ask: How do waves originate? What properties must a me-
dium possess in order to be capable of transmitting waves? What
is the mechanism by which they are propagated?

288. Water Waves. We are all familiar with water waves;
for who has not amused himself by throwing stones into still
water and watching the charmingly symmetrical figures produced
on its surface? There are few of us who are not in possession of
some vivid mental pictures that may aid us in studying this most
fascinating portion of our subject.

9. Origin of Waves. If we throw a stone into a pond and
watch closely when it strikes the water, we notice that its first
effect is to push aside the water at the place where it falls. This
action lowers the level of the water surface, thus leaving a hollow
behind the stone as it sinks from sight. Since the free surface of
a liquid always strives, as it were, to remain level, the water, which
has been thus thrust aside, rushes back as if to fill the
hollow left by the stone. But when the particles rush in from
all sides behind the sinking stone, they acquire kinetic energy,
which carries them further than they intend. The result is that

300

WAVE MOTION

301

water is now piled up in a little heap over the place where the
stone fell. /

The water particles then hastily retrace their steps, but again
they are irresistibly carried past the position they desire to occupy,
and again a hollow is formed in the surface of the pond; but this
time it is not so deep as before. This process of piling up and
retreating is repeated several times, the elevation being less
marked each time, until the motion at the point where the stone
struck ceases altogether. A back and forth motion of this sort
is called a vibratory motion. We thus reach the conclusion
that waves originate at a point lohere a vibratory motion is forced on
the medium by some outside body,

290. Characteristics of Waves. Are the waves that spread
out on the pond different when produced by a large stone from what

Fig. 172. Waves From a Large
Stone

Fig. 173.

Waves From a
Pebble

they are when produced by a small one? On trying the experi-
ment, we find that the waves caused by a large stone are larger
than those produced by a small one. If we wish to compare them,
we must agree on a method of measuring them. In order to
appreciate what the characteristics of waves are, let us suppose
that a succession of these water Vaves is suddenly frozen solid;
the shape of the surface resembles the curve in Fig. 174. Exami-
nation of this curve shows that some parts of the wave are above
the normal level of the water, while other parts are below it. The
parts above the normal level are called- crests, those below it are
called TROUGHS. The distance of the top of the crest from the

302 PHYSICS

normal level is equal to the distance of the bottom of a trough

from the same level and is called the amplitude of the wave.

The WAVE-LENGTH is the distance between two successive crests,

or between two successive troughs.

Applying these definitions to the cases of the large and small

stone^ we see that the waves started by the large stone have both

a greater amplitude and a greater
wave length than those about

Fig. 174. Shape of a Simple Wave ^^^ ^^^^^ ^^^^ ^hus there is

a relation between the nature of the vibration that started the
waves and the characteristics of the waves, so that we can judge
of the vibration by observing the waves.

291. What Waves Can Tell Us. We have just learned that
we can form some idea of the magnitude of the disturbance. that
started the waves by noting the wave length and the amplitude of
the wave. Furthermore, we can form some idea of the direction
in which the point of disturbance lies I^y noting the direction in
which the waves are traveling. Finally, v/e can .infer something
about the nature of the disturbance from the shape of the waves.
Hence we see that waves may bring us four kinds of information
concerning the source of the vibrations, viz.: 1. The direction in
which the vxLves are traveling indicatec the direction of the source.
2. The length of the waves informs its as to the rapidity of the
vibration, 3. The amplitude of the waves tells us of the violence
of the disturbance, 4. The shape of the wave allows UrS to infer
something concerning the nature of the vibrations of the source.

Wave Motion. Another important fact may be learned
from watching the water waves. If we throw a small chip on
the surface of the pond, well out from the shore, and observe its
motion when the waves pass it, we see that the chip is not carried
along in the direction in which the waves move, but that it merely
rises and falls while the wave motion passes beneath it. Since
the chip indicates the motion of the water particles about it, we
may conclude that the water does not move forward with the wave,
but merely rises and falls as the wave passes.

WAVE MOTION

303

900 00 # 000

Fig. 175. Particle 1 has Executed i vibration

This fact leads us to an important conception as to the mechan-
ism of wave motion. Thus, let us consider a row of particles
held together by cohesion or some other elastic force (Fig. 175).
If the first particle is
displaced in a direction
perpendicular to the
rov/, the force that holds
the two together will
compel the second par-
ticle to follow. But since
the connection between

the two particles is elastic, not rigid, the second will always lag a
little behind the first. Hence, when the first particle has reached
the end of its trip, the second will not have traveled quite so far,
the third will lag a little behind the second, and so on. There-
fore the condition of the row of particles, when the first one has
rpached its position of greatest displacement, will be that shown
in Fig. 175.

Particle 1, having reached its position of greatest displace-
ment, pauses there for a brief instant and then begins to retrace
its steps. While this particle^is stationary, 2 catches up with it
and reaches its position of greatest displacement as 1 starts down-
ward. Particle 3 follows 2 in the same way, and so on. Thus we
see that the successive particles reach their positions of greatest
displacement one after another. We may say that this position
of greatest displacement is passed along from one particle to
the next. But the position of greatest displacement constitutes

the crest of the wave,
and so we get a concep-
tion of the mechanism
by which waves are
propagated along a row
of particles that are
held together by cohe-
sion, or any other elastic force. The positions of the particles
when number 1 has returned to the starting point are shown in
Fig. 176.

Fig. 176. 1 has Executed i Vibration

304

PHYSICS

Fia. 177. 1 HAS Executed J Vibration

Now, when particle 1 reaches the position from which it started,
i.e., when it has completed half a vibration, it is moving with
considerable velocity. It therefore possesses kinetic energy.

This energy will cause
it to move past its origi-
nal position and to
make an excursion on
the opposite side. Hence
it will now move down-
ward, dragging the ad-
jacent particle after it, will reach a position of greatest negative
displacement (Fig. 177), and return again to the starting point.
The positions of the particles, when this has been done, are
shown in Fig. 178. Particle 1 is now in the same condition in
which it was when it began to move. Therefore, if nothing
interferes with it, it will repeat the operation just described and
continue to do so until its energy is expended.

293. Eelative Positions of the Particles. Several important
things are apparent from this discussion. In the first place, we
note that when particle 1 has executed a complete vibration, the
other particles along the wave have not yet done so. Each has
performed only part of one vibration. Each successive particle
has executed a smaller portion of one vibration than has the par-
ticle ahead of it and a larger portion than has the one behind
it. Thus, when particle 1 has completed its first vibration, 17
is just ready to begin
moving; 13 has executed
J of a vibration; 9, J a
vibration; 5, J of a vi-
bration, and the inter-
mediate particles inter-
mediate fractions of one
vibration. It is conven-
ient to have a simple word for expressing this relation. There-
fore, the particles are said to be in different phases of vibra-
tion. The phase thus means the portion of one complete

Fig. 178. 1 has Executed One ^Vhole Vibration

WAVE MC/riON 305

viMation that any particle has executed. So we may say that par-
ticle 16 has a phase zero, 12 a phase of J vibration, 8 a phase
of i vibration, 4 a phase of J vibration, and 1 a phase of
1 vibration.

294. Wave Length. Using this term "phase," we may make
a general convenient definition of wave length; for we may say
that a wave length is the shortest distance between any two par-
ticles that are in the same phase. Thus the distance 1 to 17, 2
to 18, and so on, is a wave length.

295. Time of Vibration. It is often convenient to consider
the time it takes a particle to execute one vibration instead of
considering the numbers of vibrations per second themselves. When
a vibratory motion continues to be repeated in equal time intervals, it
is called periodic, and the time taken . by any particle in execut-
ing one vibration is called the period of that vibration. Hence, when
we are considering vibrations from this point of view, we may
speak of a phase of a quarter period, of half a period, etc.
Further, it is evident that if the source of vibration executes 10
vibrations in a second, the time it takes to execute one vibra-
tion, i.e., its period, is f ^ sec. Thus, in general, if the number
of vibrations per second is represented by n, the period is always

1 ^ *

— sec.

296. Velocity of Propagation. An important conclusion
may now be drawn from the discussion of Fig. 178; for we
note that while particle 1 has been executing one vibration, the
disturbance has traveled a distance 1 to 17. This distance is a
wave length. Hence, it is manifest that the disturbance travels
along the row just the distance of one wave length while particle
1 executes one vibration. If particle 1 executes n vibrations in
a second, how far will the disturbance travel in that second?
Evidently n wave lengths. But the distance traveled in one
second measures the velocity (c/. Art. 2). Hence, we may express
this result by saying that the velocity with which a wave travels is

306 PHYSICS

numerically equal to the prod/uct of the number of vibrations per
second and the wave length. If v represents the velocity, n the
number of vibrations, and / the wave length, then

V = nl (14)

This simple relation enables us to determine the velocity of
the waves when we Can measure the wave length and the num-
ber of vibrations per second of the source. This equation, how-
ever, does not tell us how this velocity depends on the properties
of the medium through which the waves travel. We can get a
general idea of how the properties of the medium affect the velocity
with the help of equation (4), Art. 27. For / = ma, therefore

a = — . In this case / is the elastic force acting between two

adjacent particles of the medium, m the mass of a particle, and a
the acceleration given to the particle by the force /. Since a is
proportional to /, the equation shows that if / is increased,
particle 2 will have a greater acceleration; so it will fol-
low faster after 1. For the same reason 3 will follow faster
after 2, and so on ; therefore the disturbance must travel faster
along the row of particles.

On the other hand if the density of the elastic medium is
greater, each particle will have a greater mass m. In this case
the equation , shows that the particles will have smaller, accelera-
tions, so each one will move more slowly and lag more behind the
one just ahead of it; therefore the disturbance will travel more
slowly.

The exact relation of the velocity of a wave in a medium to
these two factors, elasticity and density, has been determined
mathematically and by experiment; and it has been found that
the velocity v of waves traveling in a medium having an
elasticity e and a density d is

^=\i:

(15)

297. The Types of Waves. In the discussion thus far we
have confined our attention to waves in which the motion of the

WAVE MOTION 307

particles is perpendicular to the direction in which the waves
travel. When this is the case, the waves are said to be trans-
verse. We have also pictured the tnotion of each particle as
taking place along a straight line. These restrictions were in-
troduced in order to simplify the discussion, though neither one is
essential. Thus we may just as well have waves in which the
paths of the particles are circles or ellipses in planes perpendicular
to the direction of propagation of the wave; or we may conceive
that the particles move back and forth in the direction of propa-
gation of the waves. In this latter case the particles are alter-
nately crowded together and separated, so that we have to deal
with condensations and rarefactions of the medium instead of
with crests and troughs. Waves of this type are called longi-
tudinal. We shall leam more concerning both types of waves
in the following chapters.

298. Waves of Simple Shape. In this discussion it has been
stated that we may draw conclusions as to the nature of the vibra-
tion by a study of the shape of the wave. In Fig. 174 we have
drawn a wave of particular shape. W^hat sort of vibrations werie
executed by the body from w^hich these waves proceeded? From
the simplicity of the shape of the curve we may imagine that the
vibration must be of a simple type. Now, this type of vibration
is that executed by a pendulum, as may be easily shown, by fasten-
ing a small pencil to the bob of the pendulum and drawing a card
under it in a horizontal direction, and in such a way that
the pencil writes a trace on the card while the pendulum
is vibrating. If we do this, we find that the shape of the curve
obtained is the same as that shown in the figure. The pendulum
has thus been made to construct a graph which represents the
relation between the displacements and the corresponding times
for its own motions. Vibrations of this type are called simple
HARMONIC vibrations, and the curves that represent them
graphically are called sine curves.

Since every particle in the wave executes the same kind of
vibrations as the source does, the shape of the waves that originate
from a simple harmonic vibration will be that of a sine curve.

308

PHYSICS

FiQ. 179.

Same Period, Amplitude,
AND Phase

When we have to deal with a simple harmonic motion of one
definite period only, the corresponding waves are called homo-
geneous. All other forms of waves are complex. It is perhaps
unnecessary to remark that the ones with which we are actually
familiar are in every case complex.

299. Complex Waves. If all waves which we know in nature
are complex, why do we study simple hompgeneous waves at all?

In answer to this question, let
us consider what happens when
we h^ve two or more simple
homogeneous waves traveling
through the same medium at
the same time. The result is most easily obtained from a dia-
gram. Let us begin with the case shown in Fig. 179, and
suppose that the two simple waves have the same period, am-
plitude, and phase. The disturbance that results when these
two waves are traveling through the same medium at the
same time is obtained by adding as vectors the displacements of
the particles. The result is shown by the lower curve in the
figure. We note that the resultant is a wave of the same period
and phase, but with twice the amplitude.

Repeating this operation for two waves that have the same
period and amplitude, but differ in phase by one-half period, we
get the result shown in Fig. 180. This result shows that two such
waves may be traveling in the

same medium without giving
any external sign of their pres-
ence.

If we were to add together

Fig. 180. Same Period and Amplitude,
Opposite Phases

two waves of the same period, but differing in phase by J of a
period, or by any other fraction of a period, we should find that
in every case the resultant w^ave has the same form as the con-
stituent waves. Hence we may conclude that the addition of any
number of simple homogeneous waves of a given period always
gives a resultant which is also a simple homogeneous wave of the
same period.

WAVE MOTION

sod

Fig. 181. Addition of Waves of Different
Periods

300. Wav«s of Different Shapes. If the addition of simple waves
of the same period always gives as a result a simple wave, how may
we produce waves of

complex fonn? Let
us see what the effect
will be if we add to-
gether two waves that
have different periods.
Take, for example, the
two waves drawn in
Fig. 181, one of which
is twice as long as the
other and has twice
the amplitude. We
note that the resultant
R obtained by adding them together is a wave differing entirely
in shape from the two component waves. Let us now add to
this resultant a third simple wave, with period and amplitude each
one-third of the first period and amplitude. The result is shown
at R\ The resultant obtained by adding together waves
whose periods and amplitudes have the ratios 1, J, ^ is shown
in Fig. 182. The meanings of the curves will be perfectly clear
on careful inspection.

A study of these curves will make it apparent that we
can produce waves differing greatly from one another in
shape by adding together simple homogeneous waves which

differ from one an-
other in period as
well as in amplitude
and phase. That the
number of shapes
which may be pro-
duced by the addition
of such waves must
be indefinitely great,
may be realized by any one who considers that we have at our
disposal, first, an indefinite number of possible periods; second.

Fio. 182. Another Complex Wave

310 PHYSICS

an indefinite number of possible amplitudes; and, third, an in-
definite number of possible phases.

But even if the number of shapes that can be so artificially
built up is practically without limit, is the converse proposition
true, viz., that every shape that actually occurs in nature can be
resolved into a series of simple homogeneous waves differing from
one another in period, amplitude, and phase? This problem
occupied the attention of mathematicians and physicists for a
century and a half before its final solution was reached. That
solution proved conclusively that this converse proposition is true,
and shows us how to proceed in order to separate the compound
wave into its component simple homogeneous waves. Hence, the
importance of studying the nature of the simple waves becomes
manifest, for it follows that all waves with which we are familiar in
nature are built up of these simple homogeneous waves. The study
of SIMPLE HARMONIC MOTION, which gives risc to these simple
homogeneous waves will be deferred to the next chapter.

301. Stationary Waves. There is still another kind of waves
which we have not yet discussed, but which, nevertheless,
merits attention. A jumping rope, one end of which is fastened
to a tree, must be turned at a certain rate in order to swing properly.
It is a matter of common observation that by turning it faster the
rope may be made to break into two equal loops separated by a
point where the rope moves very little. If the hand is turned
still faster and the rope is long enough, the rope may be made to
vibrate in three, four, or more parts. How are these loops
formed, and why does the rope stay nearly still in certain places?

An analysis of the operation will give us the answer. When
the hand is moved periodically it impresses a certiiin vibration
on the rope. The rope may be looked upon as a row of particles
held together by elastic forces, and so the vibratory motion of
the hand is propagated along the rope in the form of a wave, until
it reaches the other end. What happens then? Does the wave
give up all its energy to the tree, or is part of that energy reflected
so as to travel back along the rope? You may easily show, by
holding the rope rather tight and hitting it surldenly, that the

WAVE MOTION 311

wave is reflected ; for the hump raised on the rope by the stroke
may be seen to travel to the far end of the rope and then to turn
around and come back. Therefore, when you send a series of
impulses along the rope they travel in the form of waves, are
reflected at the further end, and return. We see, then, that if the
series of impulses be continued, we shall soon have generated two
trains of waves, the direct and the reflected, traveling along the
rope in opposite directions. We may infer that, when this is the
case, the result is the peculiar vibration that we get on the rope.
Let us see if this is so.

In the case we are considering, the two trains of waves have
the same period and nearly the same amplitude, and are trav-
eling along the rope at the same rate but in opposite directions,
^hey are represented by curves A and B, Fig. 183, A moving to
the right and B to the left. When the waves are in the positions
indicated at V and D in the diagram, the resultant obtained by
adding their displacements is shown by the thick black line.

If now we conceive each of the two trains of waves to have
advanced J wave length, A to the right and B to the left, their
respective positions are those shown at W and E in the diagram;
and the resultant will be the black line WE.

Two more advances, of J wave length each, bring the two into
the position shown at XyF and Y,G respectively. Clearly the re-
sultants will be as there shown. A final advance of J wave
brings the two into a position similar to their original positions, so
that the resultant for Z^H is the same as that for V,D.

W^hen these five resultants are superposed, we obtain the curves
shown at RyNyC. The similarity between this figure and the rope
in the case under consideration will at once be noted, for certain of
the points never leave their positions of equilibrium, while others
execute vibrations of greater amplitude. The positions of no
amplitude are called nodes, while those of great amplitude are
called LOOPS. Since the nodes remain at rest -with respect both
to vibratory motion and also to the motion of propagation, such
waves are called stationary waves.

When we analyze the motion of the particles in these waves
we see (1) that all the particles that move are in their positions

312

PHYSICS

of greatest amplitude at the same instant; and (2) that they are
all in their positions of equilibrium at the same instant as shown
by the straight line RNC. We note further (3) that all the par-
ticles in one loop are in the same phase at the same time, but that
their respective amplitudes are different, for the particles at the

Fig. 183. Formation of Stationary Waves

middle of the loop have a large amplitude, while those near the
nodes have a small amplitude. Another fact worthy of remark
is (4) that the particles in one loop have a phase that is differ-
ent by half a period from that of the particles in either of the
adjacent loops. And finally, we see (5) that the distance be--
tween two nodes is half a wave length. Thus it becomes
Qlear that sii^h stationary waves are actually produced by two

WAVE MOTION 313

trains of waves of the same period traveling along the same row
of particles in opposite directions. The relations between the pe-
riod of vibration and the length of the loop will be taken up in
a later chapter, for these stationary waves play a conspicuous part
in the pheno^nena of sound, and so we shall have to discuss them
further when considering that subject.

SUMMARY

1. Waves originate at a vibrating body.

2. Waves bring us four kinds of infonnation: 1. As to the
direction of the source. 2. As to the period of the vibrations.
3. As to their amplitude. 4. As to their complexity.

3. The characteristics of waves are direction of propagation,
length, amplitude, and shape.

4. A suitable medium is necessary for the transmission of
waves.

5. The particles of a medium do not partake of the progres-
sive motion of a wave, but merely vibrate about their positions of
equilibrium.

6. In progressive waves the successive particles are in different
phases at the same time.

7. Waves may be transverse or longitudinal.

8. The velocity of propagation of waves is equal to the number
of vibrations multiplied by the wave length, {v = nl.)

9. The velocity of waves in an elastic medium is equal to the

square root of the elasticity divided by the density, v = ^—,

10. When waves are superposed, the resultant is the algebraic
sum of the components.

11. The simplest kind of wave is the simple homogeneous or
sine wave.

12. The vibrations that produce these simple waves are called
simple harmonic vibrations.

13. Waves of complex form result from superposition of
simple homogeneous waves of different periods, amplitudes, and
phases.

314 PHYSICS

14. Waves of complex form may always be analyzed into a
series of simple homogeneous waves.

15. Stationary waves are produced when two trains of waves
of equal period, but traveling in opposite directions, are super-
posed.

QUESTIONS

1. Describe the motions of water at the point where a stone is
dropped into it.

2. What sort of information is derived frorii each of the four chief
characteristics of waves?

3. Describe the mechanism of wave propagation. Do the vibrat-
ing particles partake of the progressive motion of the waves? j

4. What two types of waves may we have? What is the distinctive
feature of each?

5. What can you say of the relative phases of the successive vibrat-
ing particles along a progressive wave?

6. How may wave length be defined with reference to phase?

7. Does it seem reasonable to suppose that waves can travel
through empty space?

8. What is a simple homogeneous wave? From what kind of
vibration does it originate?

9. When two or more waves are traveling at the. same time in the
same medium, how do we find the resultant wave?

10. What sort of vibrations does a pendulum execute? How may
we obtain a graph to show this?

11. Explain how stationary waves are produced?

12. What are nodes and loops?

13. How is the distance between two adjacent nodes related to
the wave length?

PROBLEMS

1. What is the period of vibration of an oarsman who makes 40
strokes per minute? Of a swhig that makes one complete oscillation
in 4 sec? In 6 sec? Of a wagon seat that makes two complete vibra-
tions in 1 sec?

2. How many complete vibrations per sec does a tuning fork make,
if its period is 2 Jo sec? What is the period of the waves started by a
paddle-wheel that has 6 paddles, and makes 20 revolutions per minute?
If these waves are 2 ft. long, what is tlioir velocity?

3. Air and hydrogen have the same elasticity, under given eon-
ditons, but air is 14.5 times as dense as hydrogen. Supposing sound
to be a wave motion, ought it to travel faster or slower in liydrogen
than in air? How many times?

WAVE MOTION 315

4. What is the length of the waves that are traversing a jumping
rope 30 ft. long when it is vibrating in 1 loop? In 2? In 3? In 4?
In 5? If the period is 1 sec when it is vibrating in 1 loop, what is the
period in each of the cases just supposed? What is the velocity of
each of the waves?

5. Plot the following cases of simple homogeneous waves travel-
ing in the same direction: 1. Two waves of equal period, amplitude,
and phase. 2. Two waves of equal amplitude and period, but of
opposite phai&e. 3. Two waves of equal amplitude and period, but
with a difference of phase of J period.

6. Plot the following cases of simple homogeneous waves travel-
ing in the same direction: 1. Two waves of equal phase, one having
half the amplitude and half the wave length of the other. 2. Combine
this resultant with a third wave whose length and amplitude are each
J that of the first wave.

7. Were you ever out in a boat when the waves were running fairly
high, or have you ever floated on your back among them? In that
case you had an excellent opportunity to observe carefully the kind of
motion that the water particles in the wave have. Did the water
move you up and down only, or was there compounded with this up
and down vibration another that was nearly horizontal? What was the
resultant path or orbit of the water particles which were carrying you
with them as they oscillated?

8. Suggest a way for producing transverse waves in a long rubber
tube. How may you produce longitudinal waves in it? Can water
react elastically to forces that tend to compress it longitudinally as well
as to forces that tend to displace it laterally? Do air particles cling
together as water particles do? Can air resist both transverse and
longitudinal stresses? If not both, which?

9. In which direction (longitudinal or transverse) does a rubber
tube offer the greatest elastic resistance, to a force producing a given
displacement? Which kind of waves then (longitudinal or transverse)
will travel faster in the rubber? Answer the same questions for
Water, wood, brass.

SUGGESTIONS TO STUDENTS

1. Throw stones of different sizes into a pool of water, and note the
differences in the lengths and amplitudes of the corresponding waves.

2. By moving both hands up and down with a regular period in a
tub of water, see if you can produce stationary waves. Keep the
hands a foot or £wo apart, and gradually change the period till the
nodes and loops are seen at definite places in the water.

3. With two companions and a kodak, go to a pond or lake and

316 PHYSICS

make a similar experiment. Instead of your hands use as sources of
waves two long poles just alike, having nailed to them equal circular
pieces of board. These will be easy to keep vibrating with equal
periods. When two of you have practiced so that you can maintain
the stationary waves, let the photographer of the party take a snap
shot of the waves. By shortening one of the poles and reducing the
size of the circular board on it, see if you can succeed in getting
waves of forms that are more complex, but nevertheless definite.

4. For many beautiful and simple home experiments on wave
motion, read Prof. A. M. Mayer's Sound (Appleton, New York). By
all means read Prof. J. H. Fleming's Waves and Ripples (E. & J. B.
Young & Co., New York).

This is a series of experimental lectures delivered to young people
at the Royal Institution, London (where Davy, Faraday, and Tyndall
worked). It is a most fascinating account of waves in water, air, and
ether.

5. If you are in the manual training class or have a shop of your
own, get together with some of your classmates and make for the
school a Kelvin wave model as described in Michelson's Ldght Waves
and their Uses (University of Chicago Press) pp. 5 and 6. Read also
pp. 1-13.

6. Another excellent wave motion model which you can easily
make is described in Jones's Heat, Lights and Sounds p. 238. Read
also pp. 236-241.

7. Let a companion hold one end of a clothes line while you hold
the other, and with it produce the result predicted theoretically by
the diagrams in Fig. 183. Let another companion who is a photog-
rapher try to get some snap shots of the line while vibrating in 1, 2,
3, 4, or more loops, and make lantern slides of them for the school
collection.

CHAPTER XV

SIMPLE HARMONIC MOTION

Note. — The authors recommend that this chapter be used only for
informal discussion on the first reading. If time is short it may be
omitted altogether.

302. Relation to Uniform Circniar Motion. The study of
simple harmonic motion is made much easier by first consider-
ing the relation that exists between this type of motion and
uniform motion in a circle. In
order to make this relation clear,
let us conceive that we have a
small body traveling with uniform
velocity v, in the horizontal cir-
cular path ABCD, Fig. 184. To
one looking at this motion from
above, the path of the body is
seen to be circular and its velocity
uniform. But if the motion be
observed from a point in the
plane of the circular path, and
at a distance from the circle in
the direction CA, the body will appear to be moving back and
forth along the straight line BD, and its motion will no longer
appear uniform. Thus, when the body is passing the points B
and D in its circular path, it will appear to be at rest. On the
other hand, when it is passing the points A and C, it will appear
to be moving with a speed which is the same as its uniform
speed V in the circular path. At intermediate points its speed
will appear to vary between these two limits, i.e., between
and V,

When we observe the uniform motion of the particle in this
latter way, it appears at each instant as if it were projected on the

817

Fig. 184. Simple Harmonic Motion

318 PHYSICS

straight line BD, ix., as if it were at the foot of a perpendimdar
dravm from it to BD, So we can readily understand how the
same effect would be produced on the distant observer if we
replace the body moving uniformly in the circular path by
an equal body that moves back and forth along the line ED
in such a way that the position of this second body at any
instant is the projection on the line BD of the position of the
first body at the same instant. Let us conceive this to be done.
It then remains for us to consider what forces must be applied
to the body moving in this way along BD, in order to produce the
required motion. The problem is not. so diflScult as at first sight
it may appear.

303. Force and Displacement. Suppose the first body is at

any point E (Fig. 184) of its circular path. The second body

must then be at F, the projection of E on BD, What force is

acting on E at this time? We have learned in Chapter V that the

force is directed toward the center of the circle, i.e., along EO,

TH/V
and is numerically equal to , in which m is the mass of the

body, V the uniform velocity in the circumference, and r the radius
of the circle. Let the magnitude and direction of this force be
represented by the vector EM. We may now conceive this
force to be resolved into two components, one in the direction
EF, perpendicular to BD, and the other in the direction
EH, parallel to BD, These components will then be rep-
resented by EP and EN, It is clear that the component
EP has no effect on the motion of the body at E in the
direction BD, It is also clear that if we allow a force equal
to the component EN to act on an equal mass at F, the motion
produced along BD will be the same as the motion in the direc-
tion BD of the body at E, Hence, the force that must be applied
to the second body at F, in order that it may always be at the
projection of E on BD, is represented by this component EN,
But what is the value of this component? From the similar

triangles, EMN and EOH, g = gor. EiVT = ^^ >< |g-

SIMPLE HARMONIC MOTION 319

But EO is the radius v of the circle and EH = FO is the distance
of the second body from 0. If this distance be denoted by d,

EN = EM X — , but EM = — , therefore substituting this value

TfVu d

for EM we have EN = — X -. But EA'' represents the force •

r r ^

that must be applied to the body at F in order to make it move
in the required manner. If we denote this force by /, we have
finally

In a given case m, v, and r are constant, therefore the force that
must be applied at each instant to a body in order to make it
vibrate in the required manner, is proportional to the distance of
the body from the center of its swing. If we call this distance the
DISPLACEMENT, and define the motion as simple harmonic
MOTION, We reach the conclusion that when a body is vibrating
in simple harmonic motion, the force at any instant is propor-
tional to the corresponding displacement,

304. The Sine Curve. Since the value of the ratio — depends

on the size of the angle EOH, it is possible to express
the force / in terms of this angle instead of the displacement. To
do this, we name this ratio the sine of the angle EOH, i.e., we
define the sine of an angle as the ratio in a right triangle of the
side opposite the angle to the hypothenuse. It is for this reason,
and also because the simple homogeneous waves considered in
the last chapter are produced by this simple harmonic motion,
that the graphs representing their shapes are called sine curves.
The abscissas of the sine curve represent the angles and the ordi-
nates represent the values of the corresponding sines,
#

305. Illustrations. We can now realize why simple harmonic
motion is of so great importance in science, for vibrations are
usually produced by elastic forces, and these are proportional to
the displacements. One of the simplest cases of such vibration is
that of a weight on the end of a spiral spring, Fig. 185. In this

320

PHYSICS

case we may easily prove that the force is proportional to the
displacement; for when we hang a weight of 100 gm on the end
of the spring and measure the displacement produced, and then
repeat the operation with a weight of 200
gm, we find that the displacement in the
second case is twice what it is in the first,
and so on.

306. Period. One other point remains
for consideration: How does the time of
vibration depend on the mass of the vibrat-
ing body and the force acting to bring it
back to its position of equilibrium? We
may find the answer to this question as
follows: When body 2 which moves along
the diameter BD, Fig. 184, is passing the
point O of its swing, it is evident that its
velocity is the same as that of body 1 at
A ; i.e., the velocity is v. But if T repre-
sent the timie it takes body 1 to travel
once around its circular path, i.e., if T
represent the period, this velocity will be equal to the circum-

2'7rr
ference of the circle divided by T, i.e., v = "V^* ^^* since at

2'jrr
the velocity of body 2 is also v, or —pp—, its kinetic energy at

this point is found by substituting this value in equation (6), Art.
39, i. e.;

^^~2 ~^\Y) ~~2r''

When, however, body 2 reaches the point B, its velocity is
reduced to 0. Hence, at B it has no kinetic energy, but this
energy has been converted into potential energy. This potential
energy, as we learned in Chapter II, is equal to the work done in
bringing the body to the position B, Now, this work is the force
/ X the displacement. So if we let F represent the force acting on
body 2 to bring it back to O when its displacement is 1 cm, the

Fig. 185. Force is Pro-
portional TO Dis

PLACEMENT

SIMPLE HARMONIC MOTION 321

force / acting at B to cause its return will be Fr, because it is
proportional to the displacement OB, and OB is r cm from 0.
Now, this force increases from the value to Fr as the displace-
ment increase^ from to r. Therefore, in order to get the work
done in moving the body from to 5, we may assume that this
variable force is replaced by a constant one. The numerical
value of the constant force that will do the same amount of work
in this case is the average of the forces at and at B; i.e., this

+ Fr Fr
force is equal to — - — = — . Therefore, the work done in mov-

Fr
ing the body from to J5 is this force — multiplied by the dis-

Fr^
placement r, i.e., W = -^. This work is equal in value to the

potential energy of the body at B; and this potential energy
is, as just stated, equal to the kinetic energy at 0. Therefore,

4w^mr* Ff^ r^ . . .,. .. « ,r» ^ ^9^ rm

—K?p2 = ~^' Solvmg this equation for !P, we have Air — = r*,

and finally /~~"

T = 2.^|, (16)

i.e., the time taken in executing one complete vihration is equal to
2ir multiplied by the square root of the quotient obtained by di-
viding the mass of the body by the force necessary to displace it
1 cm from its position of equilibrium. This force per cm is
called the force constant of the system.

It is easy to see that the expression on the right-hand side of
the equation represents a time, for its symbol in tenns of gm, cm,
and sec evidently is the square root of gm divided by that for
dynes per cm, or

g = V sec^ = seCk
gm cm

sec cm

307. Pendulum. There are several other important cases in
which this relation can be applied, besides that of a spiral spring.
Most important among these is that of the pendulum. We will now
take up the consideration of this case. Suppose our pendulum

PHYSICS

consists of a ball of lead A, of mass m, suspended on the end of a.
wire, Fig. 186. Let the distance between the point of suspension
and the center of gravity of the ball be denoted by I. The
mass in this case is clearly m, but what is the force constant?

As is well known, the pendulum
when set in motion moves back-
ward and forward along the arc
ABC, while its motion, to be
strictly simple harmonic, must
be along the straight path AC.
But if the arc ABC is small
compared with the length Z, the
difference between this arc and
its chord AC becomes so small
that we may, without apprecia-
ble error, consider that the actual
path ABC is equal to the chord
AC, and that the motion is har-
monic. The displacement of the
mass is, then, the distance AD
in the figure. The question
therefore is. How great a force
is acting on the mass m when this
N displacement equals 1 cm? To
Fig. 186. The Pendulum Diagram answer this question, conceive

the pendulum swung to the posi-
tion A, in which its displacement equals 1 cm. The only force
involved is the weight of the ball acting vertically downward. Let
the vector AM represent this force. We now conceive this force
to be resolved into two components, AN and AP, one in the
direction of the wire OA and the other perpendicular to it. The
component AN merely causes a tension in the wire, while the
other component AP produces the motion along the arc AB.
What, then, is the value of the component ^P?

Since the triangles AMP and ADD are similar (Why?) we

have

A^ ^AD
AM ~ OA'

therefore, AP =

AM^^D
OA

but AM repre-

SIMPLE HARMONIC MOTION 323

sents the weight of the body, i.e., mg, AD = I cm, and OA

= /, therefore AP = -^. This is the force corresponding to

unit displacement, therefore it is the force constant for this sys-
tem. When we have substituted this expression for the force
constant F in our equation (16), the result is

mq \ n

We thus reach the conclusion that the time a pendulum takes
to execute one complete vibration is numerically equxil to 2ir mul-
tiplied by the square root of the quotient obtained by dividing the
length of the pendulum by the acceleration of gravity. It must
not be forgotten, however, that this is strictly true only when the
displacement is so small that the chord ADC and the arc ABC
are sensibly equal and when the, mass of the wire is inconsid-
erable.

308. Uses of the Pendulum. This is one of the most important
relations in physics, for it furnishes us with a simple and very
accurate method of determining g, the acceleration of gravity.
Thus, if we solve this equation for g, we get

9 ~ rp2 f

and since we can easily measure the length of a pendulum and
ahso its time of vibration, we obtain the value of g immediately.

This equation is also of use in proving with great accuracy
that g is the same for all bodies at a given place; for if we make
a series of pendulums all of the same length, but whose bobs are
made of different substances, and if we find that they all
vibrate in the- same time, we must conclude that at a given place
g is the same for all masses. This experiifient was performed by
Newton, and later with greatest accuracy by Bessell, and the
results show that all pendulums of the same length, no matter
of what substance they are made, vibrate at a given place in the

324

PHYSICS

same time. Therefore we are justified in comparing masses by
comparing their weights, as stated in Chapter II.

Furthermore, since all pendulums of the same length vibrate
at a given place in the same time, the pendulum furnishes a most
ccmvcnient method of measuring time. The enormous importance

of the pendulum to

mankind in this respect
is so familiar, that we
need do no more than
call attention to it.

In the case of the
pendulum, it is custom-
ary to call its period
the time taken in mov-
ing from one end of its
swing to the other, not
the time taken to com-
plete a whole vibra-
tion. But since this
time is half that re-
quired for a complete
swing, the equation for
the pendulum is usually
written :

-Nfl-

(17)

Fig. 187. The Proof that the Earth Rotates

309. The Foucault
Pendulum. The 'pen-
dulum also furnishes
The experiment that
He sus-

o means of proving that the earth rotates.

shows this was first performed by Foucault in 1851

pended a ball of lead, having a mass of 28 Kg, on a steel wire

67 m long in the dome of the Pantheon in Paris, Fig. 187. On

starting the pendulum into vibration, it was found that the plane

in which it swung turned with reference to the building, and the

amount of this turning could be measured on the large circle

SIMPLE HARMONIC MOTION 325

beneath the bob, for the pendulum at each vibration would knock
down parts of a ring of sand which had been piled up around the
circumference.

The explanation of this phenomenon is as follows. On
account of its inertia the pendulum swings in a plane that has a
fixed direction in space; and, therefore, as the earth turns with
reference to this fixed plaije, this plane appears to turn with refer-
ence to the earth. If such a pendulum were suspended directly
over the north or the south pole of the earth, its plane of vibration
would turn once around in 24 hours. On the equator its plane
would not turn at all, and in intermediate latitudes it would
oscillate back and forth every day through an angle that depends
on the latitude.

SUMMARY

1. When a body moves with simple harmonic motion, the
force that acts to return it to its position of equilibrium is
proportional to the displacement of the body from that position.

2. Elastic forces are proportional to the displacement.

3. A body vibrating under the action of elastic forces executes
simple harmonic motion.

4. The periodic time of a body executing simple harmonic
motion is equal to 2 w times the square root of the mass divided
by the force constant.

5. A pendulum when its displacement is small vibrates in
simple harmonic motion.

6. The periodic time of a pendulum for a single swing is

equal to tt multiplied by the square root of — .

7. The pendulum furnishes the most accurate method of
determining g,

8. With the pendulum we may prove that g is the same for
ail masses at a given place.

9. The pendulum is our best measurer of time.

10. The pendulum furnishes us with a means of proving that
the earth is rotating.

326 PHYSICS

QUESTIONS

1. What relation exists between forces and displacements when a
body moves with simple harmonic motion? How is this proved?

2. Do bodies vibrating* under the action of elastic forces execute
simple harmonic motion? Why?

3. How does the periodic time of a body vibrating with simple
harmonic motion depend on the mass of the body?

4. What is the force constant? How does the periodic time depend
on it?

5. Upon what two factors does the periodic time of a pendulum
depend?

6. How may the pendulum be used: 1, to measure g; 2, to prove
that we may compare masses by comparing weights; 3, to measure
time; 4, to show that the earth rotates?

PROBLEMS

1. Draw a circle 6 cm in diameter: Beginning at the point cor-
responding to A, Fig. 184, divide the circumference into 12 equal parts,
and draw perpendiculars from the end of each of these arcs to the diam-
eter A C. Plot a graph in which the abscissas represent the lengths
of the arcs, measured from the point A, and the ordinates are the
lengths of the corresponding perpendiculars. Does the curve obtained
resemble that of Fig. 174? Repeat the construction, using the same
scales, but with circles 3 cm and 2 cm in diameter, taking care to have
the origin of coordinates for all the curves fall on the same vertical
line. Graphically add the three curves together. Does the resultant
resemble the curve R' in Fig. 181?

2. Draw a circle of 5 cm radius. From the point corresponding to
A, Fig. 184, lay off arcs of 15°, 30°, 45°, 60°, 75°; and drop a perpen-
dicular from each point thus determined to the diameter A C. Measure
the lengths of these perpendiculars in cm, divide the numbers that
represent those lengths by 5, and compare the numbers thus obtained
with those given in a table of natural sines opposite the same degree
number, i e., 15, 30, etc. May the sine of an angle be defined as the
lengtli of such a perpendicular in a circle whose radius is unity? Com-
pare Arts. 298 and 304 and problem 1 and see if you can understand
the meaning of the term sine curve.

3. What is the length in cm of a pendulum that beats seconds at sea
level in New York? How long must ^ pendulum be in order to vibrate
in 2 sec? In 3 sec? Write an equation expressing the relation
between the times t and t' of two pendulums and their lengths I and Z'.

4. The spring Fig. 185 is elongated 5 cm by a 100 gm weight.
What is its force constant? If amass of 250 gm is suspended on the
spring and set vibrating up and down, wliat will be its period?

SIMPLE HARMONIC MOTION 327

5. A circular brass disc is rigidly fastened to a stiff steel wire pass-
ing through its center and perpendicular to its surface. The wire is
held vertically and its upper end firmly clamped so that the disc hangs
in a horizontal plane. If the disc be turned through a small angle
^bout the wire as an axis, and then released, it will execute rotary
or torsional vibrations. With the help of Art. 86, tell what will in
this case correspond to the force constant. What must take the place
of the mass in the equation of vibratory motion in Art. 306? What,
then, is the equation for determining the period of vibration?

6. A moment of force whose numerical value is 237 X 10''' is required
to twist the disc of problem 5 through an angle of 1 radian. If the
moment of inertia of the disc has the value of 6 X 10^, what is the
time of vibration?

7. A disc suspended as in problem 5. has a diameter of 20 cm; a
force of 10® dynes, acting tangentially at each end of a diameter, is
required to give it an angular displacement of 1 radian. If its time
of vibration is 1.5 sec, what is the value of its moment of inertia?

8. A bar magnet mounted on a pivot, like a compass needle, and
deflected through an angle of 1 radian from the magnetic meridian,
tends with a moment of force whose value is 990 to return to that
meridian. If the moment of inertia of the magnet, has the value 400,
what will be the period of oscillation of the magnet when it is released?

9. The moment of force of the bar magnet in problem 8 depends
on the mutual action between the magnetic fields of the magnet and of
the earth. If the strength of the magnet's field remains constant,
how will the period of oscillation of the magnet be changed if the
strength of the earth's field is doubled?

SUGGESTIONS TO STUDENTS

1. See how nearly you can determine your own mass by swinging
in a swing, determining with your watch your period of vibration,
and getting a friend to measure with a spring Ibalance the number of
dynes necessary to pull you in the swing a measured number of centi-
meters from the position of equilibrium. If the spring balance is
graduated in pounds, remember that 1 pound-force =445,000 dynes.

2. Tie both ends of a rope to the branch of a tree about 10 ft. from
the ground, so that the rope hangs in a V whose point is about 5 ft.
from the ground. From the point of the V suspend by a single cord
a tin can, so that it almost touches the ground. Punch a small hole
in the bottom of the can, fill it with water, feet it to swinging, and see
what sort of curves the water will draw on the ground. Similar
experiments are described in Mayer, On Sound (Appleton, New York).

CHAPTER XVI
SOUND

810. Sources of Sounds. Of all the phenomena of nature
none, perhaps, is better known or more universally recognized
than the fact that sound always originates at some vibrating body.
Even an infant knows that he must shake his rattle to make it
sound, and the vibrations of a bell or drum, when they are sound-
ing, are easily felt.

Other familiar facts concerning sound are the following:
1. We can in some way tell in what direction the source of sound
lies. 2. We recognize differences in the pitch of sounds, some
seeming high and shrill, others low and deep. 3. We recognize
differences in the intensity of sounds, some being loud and strong,
others soft and weak. 4. We recognize differences in the quali-
ties of sounds, i.e., we are able to distinguish at once between the
tone of a violin and that of a piano, and can even recognize one
another by our voices in the dark or over a telephone.

311. Sound a Wave Motion. With these facts clearly before
us, many interesting questions arise. How does sound get from
the vibrating body to us? How do the vibrations of sounds of
different pitch differ? What governs the intensity of sound?
What characteristics of the vibrations of the source correspond
to the differences in tone quality? Let us proceed to find the
answers to these questions.

The first conclusion that we draw from the facts just men-
tioned is that the information which sound brings us concerning a
sounding body might be gained from a wave motion. Hence it
seems plausible to assume as a hypothesis, that sound travels from
its source to us by a wave motion. But if it is a wave motion,what
is the medium in which it travels? If we place an alarm clock
under the receiver of an air pump, we find that the alarm is no

828

SOUND 329

longer audible when the air is pumped out. This experiment
proves that ordinarily air is necessary for the propagation of
sound, and if we conclude that sound is a wave motion in air,
it lends added weight to the conclusion that waves can not be
propagated unless a suitable medium is present to transmit them.
Other facts- point to the conclusion that sound is a wave motion
in the air; for the presence of these waves in the air may be detected
by suitable apparatus, such as membranes and sensitive flames.
Another proof of this fact is derived from the velocity with which
sound travels, for this velocity can be measured by firing a gun at
one place and noting at another, distant place the time that elapses
between seeing the flash and hearing the report. It can also
be calculated from the properties of air, for in Chapter XIV we
have learned that the velocity of waves in an elastic medium is

equal to -^ -^ and both e and d for air are capable of measure-
ment. . If the two values obtained from these two different meth-
ods agree, we are well justified in saying that sound is a wave
motion in air.

312. Sound Waves are Longitudinal. It will be interesting
to calculate the velocity of sound waves in air from the formula.
In order to do this, we must determine what we mean by elasticity,
and this necessitates our knowing what kind of wave motion
sound is. Now, in Chapter XIV we have found that waves may
be either transverse or longitudinal. We there learned that when
a medium transmits transverse waves, the forces brought into
play are those that resist a sideways displacement. Hence, since
air presents no elastic force that resists a sideways displacement,
we conclude that it can not transmit transverse waves; yet since
it offers a large elastic resistance to compression, it can transmit
longitudinal waves with a large velocity. Therefore we conclude
that the sound waves are probably longitudinal, and will proceed
on this assumption to find what their velocity is.

813. The Velocity of Sound. In order to calculate the ve-

locity of sound with the help of the equation v =^ \ T> ^^ must

^=\J'

330 PHYSICS

first consider how the elasticity of the air is measured. The elas-
ticity of any substance may be defined as the ratio of the pressure
that produces the change, to the change per cm' produced. In
the case of air, the change produced by applying pressure is a

, . , ii # 1. . pressure applied

change m volume: therefore, for air, e = —, — ^-f \

° " change per cm'' m volume.

The numerical values of these quantities may be found experi-
mentally by applying a measured pressure to air confined in a cylin-
der and measuring the corresponding changes in volume. The
results of such experiments show that, for rapid compressions

dvnes
like those of sound, a pressure of 14200 ^ is required to pro-
duce a change of 0.01 ^g in volume. Hence, for air,

. = li^= 142X10' ^^
O.Ol cm^

Since the density of air at 0° C and 76 cm atmospheric pres-

1 1 42 y 10*
sure is 0.001293, we have v ^ ^^ ^ = 33150. It is to

be noted that the numerator is — — ^ and the denominator ^-^,

cm* cm'*

1 .1 P .1 .• , . gm cm ^ cm' cm* ^, ,^

and theretore the quotient is -^^ « X = — ?. i he result

^ sec'' cm'' gm sec^

is thus- seen to have the symbol ^, as it should have if it is a

velocity.

The results of many experiments in which the velocity of sound

has been measured by the method of firing a cannon and by

other methods, show that this velocity is 33170 ^ under the

conditions of temperature and atmospheric pressure specified.

Since the calculated value agrees so well with the observed value,

we may conclude that sound waves are longitudinal waves in

air.

314. Resonance. Another striking proof of the fact that sound
is a wave motion in air may be given with a pair of tuning forks
which are tuned so that they have exactly the same periods of
vibration. If one of the forks is set into vibration, the other,
though placed at some distiince from it, will begin to vibrate, so

SOUND 331

that it can be plainly heard if the first one is stopped. It must,
therefore, have been set into vibration by the regular pulsations
of the air that are started by the first fork. The little pushes of
the successive waves are applied to it just at the proper time, so
that their sum finally produces an appreciable motion of the
second fork. Every child who has pushed a heavy person in a
swing knows how the little pushes are able to set the swing into
vibration if only they are properly timed. So with the two tuning
forks; when the two forks have the same period of vibration,
the little pushes of the air waves from the first fork reach the
second fork at just the proper intervals, and thus set it into vi-
bration.

When a body is thus set into vibration by waves of the same
period as those which it is itself capable of sending out, its vibra-
tions are said to be sympathetic, and the phenomenon is called
RESONANCE. Every body, when vibrating freely, has a definite
period of vibration peculiar to it. This period is called its nat-
ural PERIOD. The period of the waves that act on a body to set
it into vibration by resonance, is called the impressed period.
The principle of resonance, then, is generally stated as follows:
A body may he set into vibration by resonance when its natural ^period
agrees with the impressed period,

315. Effect of Temperature Changes. One further point
remains for consideration, viz.: Is the velocity affected by a
change in temperature? Evidently it is, since heating the air
expands it and thus makes its density less, and a decrease in the
value of the density rf means an increase in the value of the ve-
locity V. Therefore, sound travels faster the warmer the air
is. It is easy to show that the increase in velocity is 60 ^ for
every rise of 1° C. in temperature.

316. Noises and Musical Notes. Having thus proved that
sound is a wave motion in air, let us pass on to a study of a
vibrating body that produces sound. But before entering on
this study, it will be well to make a distinction between noises and
musical notes. For a noise is a confused jumble of sounds —

332 PHYSICS

an irregular and mixed phenomenon without definite period of
vibration, while in the case of musical notes we have definite
periods of vibration; and so the numerical relations are more uni-
form, and lend themselves better to systematic investigation.
Therefore, in what follows, we shall confine our attention solely
to the musical notes, and whenever ths word ''sound'' is hereafter
useij, a continued and regular sound of definite period, i.e., a
musical note is meant.

317. The Piano. With this limitation of the meaning of the
word, we may safely say that a piano is one of the most familiar
of all sources of sound. Let us then begin our investigation by
noting some of the features of this instrument. On opening a
piano, we find that there are inside it a number of steel wires
of varying diameters, lengths, and tensions. If we strike a key, we
observe that a small hammer flies up and strikes one or more of
these wires. We further note that the wires struck are set into
vibration, and that we hear the tone as long as this vibration
continues.

Another fact that we notice is that the long and thick wires
correspond to the lower keys on the keyboard, and emit, when
vibrating, the tones of lower pitch in the musical scale. Such
observations as these lead us to ask many questions. What
relation exists between the lengths of the strings and the pitches
of the corresponding notes? Why are there just eight notes in
an octave? What is it in the sound that enables us to distinguish
between the tones of a piano and those of a violin? Why do we
call certain combinations of notes harmonious and others dis-
cordant?

318. Pitch. As has just been stated, we notice that the long
strings in the piano are the ones that produce the tones of low
pitch. We also observe that these * long strings vibrate more
slowly than the shorter ones; i.e., they execute fewer vibrations
per second. We therefore infer that pitch is in some way related
to the number of vibrations per second. That this is really the
case may easily be proved by mounting a toothed wheel on an

SOUND

333

Fig. 188. The Syren

axis and revolving it. If we hold a card so that it strikes lightly
upon the revolving teeth, we notice that the tones produced by
the wheel are different for different speeds. Since each tooth

causes a vibration when it strikes

the card, we must conclude that
the difference in the pitch pro-
duced is due to the different
number of vibrations when the
speed of rotation changes.

Another method of proving
this same thing is this: Take
a cardboard, or thin metal disk,
and punch two or three rows
of equidistant holes around its
outer edge (Fig. 188). When we
blow on one of these rows of
holes while the disk is rotating
rapidly, we notice that a tone is produced which is different when
the number of holes in the rows is different. But since, when the
disc is rotating uniformly, a difference in the number of holes
means a difference in the number of pulses or vibrations forced
on the air each second, it appears that pitch depends on the num-
bers of vibrations per second.

319. Musical Intervals. But we can prove more than this
with these instruments; for we can show that definite simple
relations exist between the vibration numbers of the notes on
the piano. Thus, if the numbers of holes in two different rows
on the rotating disc are related as 1 to 2, we perceive that the
corresponding notes are one octave apart (from do to do); if the
numbers of holes are related as 2 to 3, we find that the two notes
are a fifth apart {do to sol). Similarly, if the notes given by the
two rows of holes are do and fa, the corresponding numbers of
holes in the rows are found to be related as 3 to 4. So it
appears that the numbers of vibrations of the notes on the piano
are related to one another by simple ratios. But before we take
up the question as to the reasons for the existence of these simple

334 PHYSICS

numerical relations among the notes of the musical scale, we
must stop to discuss briefly how the lengths and sizes of the strings
affect the number of vibrations of the tones.

320. The Laws of Strings. The laws of vibrating strings
were discovered experimentally by Mersenne (1588-1648) in
1644. These laws are merely statements of the numerical re-
lations that appear in the well known facts that, other things
being equal, the longer a string is, the more slowly it vibrates;
the thicker and denser it is, the more slowly it vibrates; and the
greater its tension, the faster it vibrates. They are: Otfier things
being eqiml, the number of vibrations per second executed by a
stretched string is:

1, inversely jyroportional to its length;

2, inversely proportional to its thickness;

3, directly proportional to the square root of its tension;

4, inversely proportional to the square root of its density.

These laws are all illustrated by the strings of musical instru-
ments. The short, thin, tightly stretched strings on the piano
are the ones that give the high notes, while those for the low notes
are longer, thicker, and not so tense. The same is also true of
the violin, the cello, the banjo, and all other stringed instruments.
In these latter instruments, the strings generally all have the
same length, and on a given string the notes of higher pitch are
produced by shortening the string by pressing the finger on it.

321. Vibrating Rods. Similar relations are found to exist
for the case of elastic rods fastened at both ends or supported
in other ways. Rods may vibrate either transversely or longi-
tudinally, and the vibration numbers are different in the two cases.
This may be shown by clamping a metal or wooden rod about 1
m long and 0.5 cm in diameter in the center and then setting it
into vibration first transversely by striking it, and then longi-
tudinally by rubbing it with a damp cloth. In the former case
the vibrations will be slow enough to count; and other things
being equal j the vibration numbers are inversely proportional to
the squares of the lengths. In the latter a note of high pitch is

SOUND 335

produced and, other things being equal, the vibration numbers
are found to be inversely as the lengths. In this case the vibra-
tions are too rapid to be seen, but they may be shown by means
of an elastic ball or button suspended so it will just touch the
end of the rod. Thus it appears that the different ways in which
rods may vibrate are many, since the number of vibrations de-
pends not only on the dimensions of the rod, but also on the way
in which it is supported and the manner in which it vibrates.

Tuning Forks. One case of vibrating rods is of great
practical importance, namely, the tuning fork. This instru-
ment is universally used as a standard of pitch. Its vibra-
tions are simple harmonic, as may readily be shown by fastening
a light wire to one of the prongs and allowing the fork to trace
its motion on a moving piece of smoked glass. The resulting
curve will be found to resemble closely a sine curve (Fig. 174).

Organ Pipes. One more class of vibrating bodies re-
mains for, consideration, namely, organ pipes. In this case the
vibrating body is a mass of air inside the pipe. This column of
air may be regarded as a rod of air and its possible vibrations
investigated, as in the case of rods. Here also the vibration
number of the note given by such a column of air is inversely
proportional to the length of the column. It varies also With the
density of the air, but in all practical cases the changes in the
density of the air, due to changes in atmospheric pressure, have
so small an effect that they may be neglected.

324. Air Columns as Eesonators. Since the air in an organ
pipe has a definite mass and shape, it must, like all other bodies,
have a natural period of vibration. Therefore, if we impress on
this air a vibration whose period agrees with its natural period,
the air will be set into vibration by resonance (cf. Art. 314).
This resonance of an air column may be shown by holding a
vibrating tuning fork of a certain pitch over the top of an open
organ pipe of the same pitch. The air in the organ pipe is set

PHYSICS

into vibration by resonance, thus strengthening the tone given by

the fork. The boxes on which tuning forks are usually mounted

are made of such a size that the natural period of the air in them

agrees with the period of the fork. When the

y^] ] fork vibrates, the air vibrates by resonance, and

thus the intensity of the tone is much increased.

We may now form some idea as to how the organ
pipe is made to operate. Air, under pressure, is
admitted to the small air chamber a, Fig. 189,
whence it is blown through the slot s in such a way
as to strike the tongue L This stream of air blow-
ing across the tongue produces a vibratory motion
of the air, and when matters are so arranged that
this vibratory motion has the same period as the
najtural period of the air in the pipe, that air re-
sponds by resonance. Toy whistles, flutes, and
some other wind instruments work in a similar man-
ner. The air in the tube of such an instrument
always acts as a resonator and serves to strengthen
the vibrations produced by blowing across an edge
I I .'t of some sort. In the case of the clarinet and sax-
1 *^^ ophone the vibrations are produced by a thin elastic
i=7lVTr5 strip of metal called a reed, and in the case of the
cornet and other horns, by the lips of the per-
former, but the resonant air column is also essen-
tial.

Organ pipes are made of definite length, and
therefore each one has a definite pitch, so that we
must have a separate pipe for each note. In the
flute, however, the length of the air column may be
varied by opening or closing holes in the tube with
the fingers; so the flute can be made to produce
various notes. In the trombone the length of the
air column" is varied by a slide, which may be pushed out or
in, thus lengthening or shortening the tube. Other wind in-
struments will be found, on examination, to operate somewhat in
a similar manner.

Fig. 189. Open
Organ Pipe

SOUND 337

325. Intensity. The pitch of a note has just been found to
depend on the vibration number of the source of sound, and
this vibration number is inversely proportional to the wave
length in the air. Thus pitch and wave length are closely
connected with each other. What in the waves corresponds to
intensity of vibration in the source? The intensity of a wave
must depend on the intensity of vibration of the source of the
wave, and this latter is greater the greater the amplitude. Thus
the amplitude of the sound waves tells us in a general way of the
intensity of vibration ,of their source. It may readily be shown
that the intensity is proportional to the square of the amplitude.

The intensity of sound from a sounding body — a stretched
string, for example — may be increased by changing the way in
which the string is mounted. Thus, if the string is stretched
between two heavy blocks of iron, the sound from it is not very
intense, because the string has a small area, and so it slips, as it
were, through the air without imparting much energy to it. But
if the string is stretched over bridges on a large thin board, the
bridges and the board are set into vibration by the string, and,
since the board has a large area, a large amount of energy is
transmitted to the air by it. Therefore the sound from the
string is louder. Such a board is called a sounding board.
The air in the tubes of organ pipes and other wind instruments
serves a like purpose, as has just been mentioned, for the large
mass of this air enables them, when set into vibration, to transfer
more of their energy of vibration to the surrounding air, and so
to increase the intensity of the sound.

SUMMARY

1. Sound originates at a vibrating body.

2. Sound gives us information concerning: 1, the direction
of the sounding body; 2, the number of vibrations per second;
3, the intensity; and 4, the nature of the vibrations.

3. Sound is a wave motion of the air.

4. Sound waves are longitudinal.

6. The velocity of sound in air is 331.7 -^- at 0° C.

338 PHYSICS

6. The velocity of sound in air increases 60 g^- for a rise in
temperature of 1° C.

7. Every elastic body has a natural period of vibrjttion.

8. A body may be set into vibration by resonance when the
impressed period is equal to the natural period of the body.

9. The pitch of a note depends upon its vibration number.

10. The vibration numbers of the notes of the piano are
related to one another by the simple ratios 1: 2: 3, etc.

11. The length of a sound wave in air depends on the vibra-
tion number of the source.

12. The vibration number of a stretched string is inversely
proportional to the length and the diameter of the string; and directly
proportional to the square root of its tension and its density.

13. The number of vibrations of an air column is inversely
proportional to the length of the column.

14. The vibrations of a tuning fork are of the simple harmonic
type.

15. The intensity of a sound wave is proportional to the square
of its amplitude.

QUESTIONS

1. What is the origin of sound? How is this known?

2. Why are we led to suppose that sound is a wave motion?

3. Why do we believe that sound consists of waves in air?

4. How do we prove that this supposition is correct?

5. How do we define the elasticity of air?

6. What leads us to conclude that sound waves are longitudinal?

7. How fast does sound travel in air at 0° C? Does its velocity
depend on the temperature?

8. Does the velocity of sound in air depend on the pressure of the
barometer?

9. Give some familiar examples of resonance, and show how the
phenomenon helps us to prove that sound is a wave motion of the air.

10. On what does the pitch of the note depend? How is this
proved? What can you say of the lengths of the waves that start
from two vibrating bodies of different pitch?

11. On what four characteristics of a string does its number of
vibrations depend? In what way does it depend on each? How are
these relations determined?

12. Are the transverse vibrations of a rod faster than the longitu-
dinal? How do you know?

SOUND 389

13. How is an organ pipe set into vibration? What is the action
of the air in it? '

14. How may we increase the intensity of the sound emitted by
a vibrating string? How is this done in the piano, the violin, the
guitar, the cornet, and the trombone?

PROBLEMS

1. Consider a stretched violin or piano string. The string rests
on supports near its ends. At these points the string is not free to
move. Are they, then, similar to that end of the jumping rope (Art.
301) which was tied to the tree? Will these points be nodes when the
string vibrates? Will it vibrate in stationary waves? If the string
vibrates as a single loop, like a jumping rope when being used to jump,
what is the relation between the length of the string and the distance
between the nodes of the stationary wave along it? How does the
distance between the nodes in a stationary wave compare with the
wave length of the wave? If L represents the length of the string,
and Zi the wave length of the wave formed when the string vibrates
in one loop, show that L = J Zj. Similarly, if I2 represents the wave
length when the string vibrates in two loops; Zg, the wave length cor-
responding to 3 loops; Z4, that corresponding to four loops; show that
L = i Zi = IZ2 = i ^3 = t ^4, etc.

2. Draw curves similar to CNR, Fig. 183, showing the appear-
ance of a string of length L vibrating in 1 loop; in 2 loops; in 3 loops;
in 4 loops.

3. Consider a solid rod 1 m long, 5 cm in diameter, and clamped
in the middle. Since the middle point is clamped, there must be a
node there when the rod is set vibrating in stationary waves. Strike
one end of the rod so as to set it vibrating transversely and watch its
motion. Do the points at the ends vibrate with greater amplitude
than those near the middle? If so, and the rod is vibrating in sta-
tionary, waves with a node in the middle, where do the centers of the
loops lie? In a stationary wave, is the distance between the centers
of the loops equal to that between the nodes? What is, then, the
relation between the length of the rod and the length of the stationary
wave on it?

4. In the case of the rod of problem 3, the distance from the end
of the rod to the node was J of the wave length of the stationary wave.
Where must the nodes lie if the ends of the rod are free and it vibrates
so that the length of the rod is 'a whole wave length? In this case,
must the middle of the rod be left free to vibrate as a loop? Draw
diagrams showing the nodes on the rod when vibrating so that its length
is i wave; f waves; f waves. If the rod is always left free at ita ends,
can the ends ever be nodal points?

340 PHYSICS

5. In the open organ pipe, Fig. 189, the air is free to move at both
ends of the pipe. When the air column vibrates in this way, where
will the node lie? What will be the longest stationary wave which the
column of air in the pipe can form? If you draw diagrams showing
the positions of the nodes when the air column is vibrating so that its
length = J wave length, j wave lengths, etc., will these diagrams
differ froni those of problem 4 in any way except that the vibrations
of the particles of air are along the pipe instead of transverse, as the
vibrations of the rod's particles are?

6. The longest wave of an open organ pipe is twice the length of
the pipe.^ Sound travels with a velocity of 1120 ^- at 15° C. What
is the length of an organ pipe that gives the tone middle c {ut^ of 256
vibrations per second?

7. The longest open pipes in church organs are 32 ft. long. What
is the pitch of the tone given by one of them?-

8. We have seen that in an open pipe the free ends are always the
places where the air particles vibrate with greatest amplitude, i.e., the
open end always corresponds to the middle of a loop. If a pipe is closed
at one end and open at the other (a stopped pipe), will the closed end
be a node? Then how does the length L of the stopped pipe compare
with the length l^ of the stationary wave? If you blow very gently
across the mouth of such a closed tube, it sounds its fundamental
or lowest tone; but if you blow harder it gives a higher tone. Does
this indicate that another node has been formed, so that the air col-
umn is vibrating in shorter stationary waves? Diagram the condi-
tion of the air when there are two nodes, one, of course, at the closed
end and the other between the two ends. If I2 now represents the
length of the stationary wave, what fraction of Z2 is L? Blowing still
harder across the open end, you may get a still higher note, which
corresponds to stationary waves when there are three nodes, including
the one at the closed ends. Diagram this condition of the air column,
and state the relation of L to Z3.

9. How long must an open pipe be in order that the longest sta-
tionary wave in it shall be equal to that in a closed pipe 20 cm long?

10. Can you compare the stationary waves on a rod, clamped at
one end and free at the other, with those of a closed organ pipe, and
show that L = \li = m = \lzy etc.? Show, by diagrams, where the
nodes ought to be. Clamp a long, flexible, and elastic rod in a vise
and see if you can make it vibrate transversely in 1, 3, and 5 half -loops.

11. The numerical value of the elasticity of water is found to be
205 X 107, its density 1 gm per cm^; what is the velocity of sound in it?

12. When a tuning fork vibrates, its center of gravity remains
at rest. How must the prongs move with reference to each other in
order that this may be true?

SOUND 341

13. When the two prongs of a tuning fork are approaching each
other while vibrating, they compress the air between them, thus start-
ing a condensation in the wave. At the same instant is the air on the
outer sides of the prongs compressed or rarified? If the fork thus
starts a condensation and a rarefaction at the same time, why do not
the two destroy each other^s effects so that we hear no sound? Hold a
vibrating tuning fork near your ear, turn it about its long axis, and
see if you can find any positions in which no sound is heard. If you
find them, explain their presence.

14. What is an echo? Suggest a way of determining approxi-
mately, with the aid of a watch, the distance of a hill which gives an echo.

15. Caii you prove by geometry that when sound spreads out
from a small source, the intensity of the energy received on one cm^ of
surface is inversely as the square of the distance? If you can do this,
explain the use of speaking tubes and megaphones.

SUGGESTIONS TO STUDENTS

1. Have you ever noticed the tones given by telegraph wires when
the wind is blowing across them? How do these tones arise? Make
an iEolian harp and put it in your window.

2. If you have a flute, measure the distance from the mouthpiece
to the hole that gives a certain tone and calculate the vibration num-
ber of the tone.

3. Perhaps you have noticed that when a rapidly moving loco-
motive is whistling as it passes you, the pitch of the whistle changes
at the instant when it reaches you. Does the pitch rise or fall while
the train approaches? While it recedes? Can you apply your knowl-
edge of the composition of motions to explain why this is so? Suspend
an electric bell by wires from 10 to 30 ft. long, connected with a battery
and push button. Swing the bell through a long arc and keep it ring-
ing. What changes occur in the pitch? Why?

4. The vibrations of organ pipes are well presented in Sedley Tay-
lor, Sound and Music (Macmillan, New York). You will also find a
great deal of interesting information about sound and music, and about
fog signals, in Tyndall On Sound (Appleton, New York).

5. For much information in very concise form, see Jones's Heatj
Lightt and Sound (Macmillan, New York). Blaserna's Sound and Music
is also good (Appleton, New York). For home experiments, see
Mayer's Sound, and Hopkins's Experimental Science.

CHAPTER XVII
THE MUSICAL SCALE

326. Development of the Musical Scale. The first impor-
tant problem concerning the musical scale is that of finding why
we have selected certain particular pitches and put them together
in a certain way to form the gamut of the piano. From the dis-
cussion in the last chapter, it appears, that within certain wide
limits, strings may be made to execute any number of vibrations ;
and, therefore, with a large number of strings differing from one
another in diameter, length, and tension, such as we have in the
piano, we are able to produce a series of tones whose vibration
numbers shall be related to one another in almost any way that
we may choose.

In the preceding chapter we proved, with the help of the
punched disc, or syren (Art. 318), that the vibration numbers of
the familiar notes, dq, fa^ sol, do, were related by the simple
ratios f , f , f . It therefore becomes of interest to try to find
out why we pick out a certain particular set of notes whose vi-
brations are related to one another in such a simple and definite
way. The answer to this question is in one way very simple,
and in another very complex; but before we can answer it, we
must find out what the relations between the numbers of vibra-
tions of the different notes of the piano scale are, i.e., we must
discover the manner in which that scale is constructed.

The history of music helps us here; for from it we learn that
mankind has not always had a musical scale, and that different
peoples select different scales. We, for example, would find it
difficult to recognize the productions of a Chinese orchestra as
music. But even nations of our own type of civilization have not
always had harmony as we now know it. The music of the early
centuries of our era sounds harsh and ofttimes discordant when
compared with modern compositions. Thus we learn that the

312

THE MUSICAL SCALE 343

present musical scale was not used in early times, and ihajt it has
gradually developed into its present form, this form having been
reached during the 16th century. Since Johann Sebastian Bach
was the first who composed masterpieces in the modem scale,
he is often called the '* father of modern music."

327. The Eelated Triads. Go to the piano and play the
two notes, middle c and g. Together they form a compound
tone that pleases us, so we call it harmony. But this combination
of c and g does not sound rich and full. We like the eflPect better
when we add the note e and play together the three notes c-e-g.
This combination of three notes satisfies us somehow; and if we
strengthen the efiFect by playing also the octave of some or all
of the three notes, we are still better pleased. Since the com-
bination of these three notes produces such an efiFect on us, we
make great use of it in musical compositions. We call this com-
bination, i.e., the combination do-^mi-sol, a major triad.

Now, although the major triad is a pleasing combination, it
becomes monotonous when played continuously. Hence," we
must seek for other triads for variety. When we try various
other triads on the piano, we find that there are two others that
seem to harmonize with the first. Thus, if we play c-e-g, g^b-d,
c-e-g, we recognize that the two triads are in some way related;
similarly, if we play c-e-g, f-a-c, c-e-g, which are recognized as
the familiar amen at the end of hjonns. If now we play the
three triads thus found in succession, viz., c-e-g, f-a-c, g-b-d,
c-e-g, we perceive not only that we have played a pleasing suc-
cession of chords, but also that we have been left with a sense of
repose. We know that the piece has ended and we are satisfied.
Therefore we conclude that in some way these three triads define
a scale or key. These three triads, which together define a scale,
are called the tonic {do-mi-soJ), the dominant {soUsi-re) and
the SUBDOMINANT (Ja-la-do) triads.

328. The Vibration Nnmbcrg. Having thus discovered that
these three triads define a scale and that we select them solely be-
cause they please us and give us a sense of harmony and repose, let

844 PHYSICS

us next find out if there are any numerical relations between the
vibration numbers of the notes that compose them. This may
be done in a number of ways, but is accomplished most easily
by taking a string of given substance and tension and finding how
its length must be changed in order to produce the tones of the
triad (c/. Art. 320). The experiment is easily performed with
a guitar, banjo, or mandolin string; for we have but to measure
the length of the string from the nut to the bridge and then meas-
ure the distance from the bridge to the frets that give the tones
mi and soL When we do this, we find that these lengths are
related to the lengths of the string by the ratios \, f, |, i.e., if
the whole string gives the note do, the note mi is given by | of
the string, and the note sol by § of that length. But since the
vibration numbers of strings are inversely proportional to the
lengths of the strings, we see that the vibration numbers of the
notes of the triad are related to one another as are the ratios {, f ,
f ; or, what amounts to the same thing, by the ratios, 1, f , }. Thus
we prove that the vibration numbers of the notes in a triad are
related to one another as are the simple ratios 1, f , J.

). The Major Scale. If we assunie that the note c exe-
cutes 24 vibrations each second, we see that the numbers of vi-
brations of the three notes of the triad c-e-g are 24 X 1 = 24,
24 X f = 30, 24 X i = 36. What will then be the vibration
numbers of the notes of the second triad g-b-dt Since the lowest
note of this triad is the same as the upper note of the other, the
vibration numbers of its notes will- clearly be 36 X 1 = 36, 36 X |
= 45, 36 X f = 54. To get the corresponding vibration lum-
bers for the triad /-a-c, we note that it contains a note c which is
an octave above the c in the triad c-e-g. But we learned in the
last chapter that the octave executes twice as many vibrations a
second as the lower note. Therefore the number of vibrations
of the upper c is 48. Since this note is the third note in this triad,
this number corresponds to f . Hence, the note / in this triad will
be 48 X I = 32, and the note a, 32 X f = 40, or 48xf =40.

We thus find the following vibration numbers for the notes
in these triads:' c = 24, e = 30, gr = 36, 6 = 45, d = 54, c = 48,

THE MUSICAL SCALE 845

a = 40, / = 32. We note that all the numbers lie between 24
and 48, excepting d, which corresponds to 54. In order to bring
this number within the desired octave, we transpose this note
down one octave and thus find the vibration number of the lower
d to be V" = 27. We now arrange these notes in the order of
their vibration numbers and get the series:

c d e f g a b c

24 27 30 32 36 40 45 48

On inspecting this series we find that it contains all the notes
of the musical scale which correspond to the white keys on the
piano, i.e., the notes do, re, mi, fa, sol, la, si, do. But this series
of notes is composed only of those notes which appear in the
three triads which we have found necessary to define a scale
or key. Hence we see that the musical scale is selected as it is,
in order that it may contain all the notes necessary for the pro-
duction of the three major triads which we have selected for the
reason that they produce, when played together, a feeling of
satisfaction and repose. It appears, then, that these particular
notes have been adopted for a musical scale because something
connected with our perception of sound leads us to pronounce
certain combinations of tones harmonious or pleasing, and others
discordant or disagreeable.

It is interesting to observe that the ratios of these numbers
can always be expressed as the ratio of small whole numbers.
This fact was clearly perceived as long ago as B.C. 525 by Pythag-
oras, who propounded the problem in the question. Why do
we call a combination of tones harmonious when the vibration
numbers of the component tones are related to one another by
the ratios of simple whole numbers? This problem of Pythag-
oras remained without answer for over two thousand years.
Helmholtz, in 1871, finally solved it. But before passing to his
solution of it, we must complete the definition of the musical
scale, for we have only found the ratios that exist among the
vibration numbers that correspond to the notes of the white keys
of the piano. We have yet to find out why there are black keys
also. Further, we must determine the actual numbers of vibra-

346 PHYSICS

tions of the diflPerent notes, for the numbers that have just
been given express merely their ratios.

330. The Complete Scale. The necessity for the black keys
becomes apparent when we wish to play a set of triads beginning
with e instead of with c. Since the vibration number of the note e
is represented by 30 in the table just given, we see, by applying
our ratios 1, f, J, that the relative vibration numbers of the
notes in the triad beginning with e would be 30 X 1 = 30, 30 X
I = 37J, 30 X f = 45. The note represented by 45 already
exists in the scale at 6, but we have no note corresponding to
37J. Since this number falls nearly half-way between 36 and 40,
it has been found necessary to add another note to our scale about
half-way between g and a. This note is called g sharp, and it
is clearly added to enable us to play scales that begin on e instead
of on c, thus increasing the number of scales that can be played
on the instrument. Similarly, if we wish to begin a scale on a,
which is represented by 40, the second note of the triad would be
represented by 40 X f == 50, or by V" = 25; and the third by
30 X f = 60, or by Y = 30. Now, the note 30 already exists
in our series, but an extra note corresponding to 25 has to be added
between 24 and 27. This note is called c sharp. Similarly, by
figuring the numbers of vibration of the triads that begin on the
notes d and 6, we find it necessary to add other notes between /
and g and between d and e. The reason for adding the black
keys is therefore apparent. They enable us to play scales that
begin on notes other than c.

But as we proceed with this addition of notes our series soon
becomes very complex. For when we construct the triad that
begins on d, we find the corresponding numbers to be 27-33i-
40^. We can supply the note represented by 33 J by adding
one between / = 32 and g =36. But the number 40^ does not
agree with a = 40, though it comes pretty near it. Similarly,
when we come to supply the triad which shall have c = 48 for
its middle note, we find the numbers 38f-48-57|, or reducing
the latter one octave 28f-38|-48. Now, we have already
added one note between 36 and 40, viz., g sharp = 37^, and

THE MUSICAL SCALE 347

this differs slightly from the one that now appears to be neces-
sary, viz., 38|. If we carry this process of working out triads
further, we find that very many notes would have to be added
in order to make it possible to play scales which begin on all the
notes of the scale of c. A keyed instrument of the piano type
would require about 70 keys to the octave and would soon become
too complicated to manage. Yet every one knows that we can
play all the different scales on the piano. How, then, is the diffi-
culty avoided?

331. Tempered Scale. The answer is simple. We insert
extra notes which do not exactly satisfy either of the required
conditions, i.e., when two notes have vibration numbers that are
very nearly equal, we take an average note and let it do for both.
Thus, instead of having a note 40 and another 40^, we make one
note do for both by tuning the strings so that the number of vi-
brations shall correspond to about 40^; similarly with the other cases.
We do not have on the piano a note 37^ and another 38|, but one
note corresponding to about 38, etc. By doing this .we do not
produce the triads in perfect tune, but we approach nearly enough
to perfect tune for all practical purposes. The scale in which
these adjustments have been made is called a tempered scale,
to distinguish it from a scale in which the notes are related by the
correct ratios.

All keyed instruments, like the piano, the organ, the clarionet,
in which each key corresponds to a note of definite pitch, must be
tuned to the tempered scale. On the other hand, stringed in-
struments, like the violoncello and the violin, may be played in
the pure scale. It is for this reason that many musicians find the
piano music disagreeable. It is related of Handel that he could
not bear to hear music played in the tempered scale, so that he
had constructed for himself an organ which had keys for every
one of the notes demanded by the theory. A musician like Handel
might be able to play upon a keyboard as complicated as this,
but less gifted individuals would evidently be able to do nothing
with it.

In tempering the scale, what method is employed? Do we

348 PHYSICS

simply guess at the probable location of the notes desired, or do
we adopt a fixed principle which shall render the departures
from accurate tuning as small and as evenly distributed as pos-
sible? Evidently the latter procedure is the only strictly scien-
tific one. The principle which is adopted is that of dividing
the interval of the octave into twelve equal parts. Since the
numbers that represent the series of notes express ratios merely,
this division into twelve equal pa,rts must be done by finding a
number such that, if we multiply 24 by it twelve times in suc-
cession, the result will be 48. This number has been found to be
1.059, and if we multiply 24 by it twelve times we get for the
numbers that correspond to the notes on the piano scale those
indicated in the following table. The numbers that indicate the
true intonation are added in order to make clear just how great the
departures of the tempered scale from the theoretically correct
one are:

Natural

Tempered

c» d\>

25.43

26.94

d# eb

28.55

e ,

30.25

32.05

f1f9^

33.96

35.98

gf# Ob

38.12

40.38

a# 61-

42.80

45.33

If we examine these numbers we see that the notes d and g
are but slightly out of tune; while some of the others, like a, are
badly so. However, for an instrument like the piano and the
organ, in which the notes are fixed, this distribution of error
seems to be the best that can be made without unduly increas-
ing the number of keys.

THE MUSICAL SCALE 349

332. Standard Fitch. One more factor remains to be deter-
mined before the scale is completely defined. The numbers that
have been given express merely the ratios between the vibration
numbers of the different notes. In order to fix the scale com-
pletely, therefore, we must state how many vibrations some par-
ticular note gives. Two definitions of this sort are in common
use. The physicist says, I will define the note middle c to be
that note which executes 256 vibrations per second. The num-
bers of vibrations of the other notes of the scale may then
be found by multiplying 256 by the ratios given in Art. 329.
Thus, the vibration number of middle g is 256 X f J = 384,
etc. The musician, however, uses a different absolute pitch,
for he defines the note a, which is the a of the violin, to be
that note whose number of vibrations is 435 per second. This
definition, viz., a =435, is called the international standard

PITCH.

333. Forced Vibrations. Having thus found what the numer-
ical relations are between the notes of the musical scale and learned
that they have been so chosen because of something connected
with our perception of sound, we will now proceed to see if we
can find out what that something is. Although our ears are
complicated structures, yet the physical principle that finds ap-
plication in their operation is rather simple. It is none other
than that of resonance, with which we have become familiar in
Art. 314.

We there learned that a strict agreement between the natural
period of the body and the impressed period of the wave is neces-
sary for the production of resonance. While this is true in many
cases, it is not always true, for light flexible objects will often
vibrate by resonance when the impressed period coincides only
approximately with the natural period. In such cases the vi-
brations will be most violent when the agreement between the
periods is exact, and Will diminish in intensity as the difference
between those periods increases. The vibrations produced by
resonance when the natural and the impressed periods are not
the same, are called forced vibrations.

350 PHYSICS

334. The Ear. The next step in solving the riddle of Pythag-
oras is indicated by the question, does the ear perceive sound
by resonance? Are there in the ear a series of bodies whose
natural periods of vibration are different, so that they would
be set vibrating by different impressed periods? The answer
to this question can, of course, be found only by an anatomical
investigation of the construction of the ear. This has been done,
and it is found that there are in the ear a large number of jfine
fibers of different lengths. These fibers are fastened at one end
to a membrane which is inside a tiny cell that looks like a small
snail shell, and is therefore called the cochlea. The cochlea is
full of liquid, so that the fine fibers — called fibers of corti,
after their discoverer — are surrounded by the liquid. The mem-
brane in wliich these fibres end is connected to the auditory
nerve which carries the sensation to the brain. The arrangement
of the cochlea and the other parts of the ear is shown in Fig.
190.

The main thing that interests the physicist in the construction
of the ear is the presence of this series of fibers of Corti; for these
may be a series of bodies which have
natural periods of vibration, and which
may, therefore, be set into vibration
by resonance by notes of different pitch.
And this is what we believe them really
to be — a veritable set of resonators,
each tuned to one of the notes which
we are able to distinguish within the
range of the musical scale. But how
many would that be? Experiment tells
us that we can not hear sounds whose numbers of vibrations are less
than about 30 per second or more than about 30,000. So these
little resonators in our ears must be tuned to notes that fall within
this range. Attempts have been made to count them, and there
are found to be about 3,000 altogether. Therefore there must
be one for each difference of about 10 vibrations. It is probable
that there are more than this in the middle of the aiusical scale
and fewer at the outside limits; but however this may be, it is

THE MUSICAL SCALE S51

certain that there is not one for each diflPerence of one vibration.
And yet we can detect difiFerences of less than this amount in
pitches and can hear tones with all conceivable numbers of vi-
brations within the range just mentioned. How is this possible
if there is only one resonator in the ear for each difference of 10
vibrations?

To account for this, Helmholtz holds that because these Corti
fibers are flexible and light, they vibrate by resonance in response
to notes whose periods are nearly the same as their own natural
periods. If this is so, then a fiber whose natural period is y^^
sec will respond to impressed periods that lie, say, between jj^
and Yki^ sec. Hence we see that any particular note must affect
several of the little resonators in the ear, acting most strongly on
that one whose natural period is nearest to the impressed period.

To sum up what we have thus far learned, we see that the ear
contains a large number of tiny resonators (fibers of Corti) which
are tuned to different notes throughout the range of audible tones;
when a sound wave of definite period falls on these resonators several
adjacent ones are set into vibration. This knowledge of the con-
struction of the eAr is essential if we are going to understand at
all the reasons for harmony and discord.

335. Beats. We may now ask what sort of excitement
of these fibers of Corti would be disagreeable. We can imagine
a sort that would probably prove disagreeable by considering the
similar case of light; for we all know well that a steady light is
necessary for any comfort in seeing, while a flickering light, pro-
vided the number of flickers is neither very great nor very small,
is intolerable. May it not be that a flickering sound would be as
intolerable as a flickering light? But what is flickering sound?
Clearly one in which periods of sound and silence follow one
another closely, just as the flickering light is one in which periods
of light and darkness follow one another closely. Do we ever
have flickering sounds? Let us see.

Take two tuning-forks, or organ pipes, or other sources of
sound whose vibration numbers differ slightly, one being, say
greater than the other by one. Conceive them to be started at

352 PHYSICS

once in opposite phases. Then the two waves (Fig. 191) which they
send out will start in opposite phases, and an observer will hear no
sound. But since one of the waves is shorter than the other,
and since they both travel with the same velocity, the phase of
the shorter wave will gradually gain on that of the longer wave
until' the phases coincide( a, Fig. 191). When this condition has

lAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAy
/\AAAAAAAAAAAAAAAAAAAAAAAAAA/N

Fig. 191. Beats

been reached, the two waves add together their effects, and a
period of loud sound results. If there is in each second one more of
the shorter waves than of the longer, the loudest sound will occur at
a, in the middle of the second. As the waves proceed further the
shorter again outstrips the other in phase, until at the end of the
second b, they are again opposite in phase and we hear no sound.
If the difference in the vibration numbers of the two notes is
2, then there will be two periods of silence in each second, and so on.
If iVj and iVj represent the numbers of vibrations per
second of the two notes, then N^ — N2 =" n will be the number of
periods of silence in a second. Two tones that produce flickering
sound in this way are $aid to give beats, and the number of
beats per second is equal to the difference in the numbers of vi-
brations of the two.

336. Discord Due to Beats. Having thus found out how a
flickering sound may be produced, let us see if the result is disa-
greeable. The experiment may be tried in the laboratory by
sounding together two organ pipes, or two tuning forks of slightly
different pitch. They Jire also readily audible when two adjacent
lower notes on the piano are sounded together. When they are
slow they can be counted. When they get faster they become
disagreeable, and when they become very rapid — more than

THE MUSICAL SCALE 353

about 30 beats per second —they fail to be distinguishable and the
disagreeable sensation ceases. Therefore Helmholtz concludes
that discord is due to beats, and that we call two tones discordant
when their combination produces between four and thirty beats
per second.

But even with this explanation of discord we are still far from
our goal. For if notes that give not over 30 beats per second are
discordant, why do we object to the combination c-/# and prefer
the harmony, c-^? Since the numbers of vibrations of c, /# and
g a*re 256, 376, and 384, the numbers of beats in these .two cases
are 120 and 128. Since these numbers of beats both fall outside
of the disagreeable limit of 30, why should we judge one of the
combinations of tones harmonious and reject the other as dis-
cordant? Before we can answer this question we shall have to
discover the reasons for differences in quality between the tones
of different musical instruments. As this inquiry is somewhat
long, we shall devote the next chapter to its study.

SUMMARY

1. The musical scale has not always existed in its present
form.

2. The notes whose vibration numbers are related by the ratios
I, f , J are called a major triad.

3. We choose the ratios f, |, J for the triad because we find
by experiment that the combination of the corresponding notes
is pleasing.

4. There are three triads whose relationship is very close, viz.,
the tonic, the dominant, and the subdominant.

5. These triads together contain all the notes of the musical
scale, and therefore define the scale.

6. The ratios of the vibration numbers for the notes of the
major scale are expressed by the following numbers:

c d e f g a b c

24 27 30 32 36 40 45 48

7. Intermediate notes have to be added to this scale if we wish
to be able to play scales beginning on notes other than c.

354 PHYSICS

8. This addition of intermediate notes makes tempering
necessary.

9. Pianos and organs are tuned to the tempered scale.

10. The physicists' standard of pitch is c = 256, while the
musicians' standard is a = 435.

11. The ear contains a series of resonators called fibers of
Corti, whose natural periods lie within the limits ^and jt^dd sec.

12. A Corti fiber is affected by a vibration even though the
agreement between its natural period and the impressed period
is not exact, so that one impressed period produces vibrations
in more than one fiber.

13. Two notes of different numbers of vibrations produce
beats. The number of beats per second Is equal to the difference
between the numbers of vibrations of the two notes.

14. When two notes produce from 4 to 30 beats per second,
the sound flickers and we call it discord.

QUESTIONS

I. How can we find the ratios of the numbers of vibrations of
the three notes in a triad? What are those ratios?

. 2. Why do we say that the tonic, dominant, and subdominant
triads are related? Why do they define a key?

3. How do we derive the relative numbers for the second and third
triads from the first? What are these numbers?

4. What is needed for defining a scale in addition to these numbers?

5. Why is it necessary to add the black keys to the piano key-
board?

6. Why do we temper the piano notes? Upon what principle is it
done?

7. Will a continuous force set a body vibrating? What sort of
force will?

8. What relation must exist between the period of the impressed
force and the natural period of a body in order to produce sustained
vibration? When this relation is not exact, can resonance occur?

9. Can a stretched string detect- a sound? If so, when?

10. What do we suppose to be the action of the fibers of Corti in
the ear when a sound is heard?

II. Do we believe that one or that more than one fiber of Corti
vibrates by resonance when a note of definite period is impressed on
the ear? State the reasons for your answer.

THE MUSICAL SCALE 355

•12. What sort of sound is disagreeable to the ear? How is such a
sound produced?

13. If two notes have relatively Ni and ^^2 vibrations per second,
how many beats will they produce when sounded together?

14. Why are beats disagreeable only when we have more than 4 or
less than 30 per sec?

PROBLEMS

1. Suppose a banjo string to be 90 cm long from the bridge to the
nut. Calculate the distances from the bridge to the frets that give
the various tones of one octave of a major scale, using the ratios of
the vibration numbers as given in Art. 329.

2. Harmonics are produced on a violin string by lightly touching
the string at points i, J, J, i, etc., of the length of the string. This
forces a node on the strirtg at the point touched, and causes the string
to vibrate in 2 loops, 3 loops, 4 loops, etc. What are the tones that
may be obtained from the string in this way?

3. Pythagoras proposed to determine the tones of a musical scale
by taking only intervals of a fifth (do-sol), starting from a given note.
Beginning with c = 24 vibrations, find the number corresponding to
its fifth, g. Then find the fifth of gr, by multiplying by the ratio |.
Continue the process, reducing eaqh number that falls outside the
limits 24-48, by i, and see if you can find out why such a scale is
impracticable. This process amounts to raising } to the power n.
Is I commensurable? Will (I)** be commensurable? Will you ever
by this process reach a note that is an octave of the one from which
yoii started?

4. Calculate the vibration numbers of the triad beginning on a = 40
and those of the one whose middle note is / = 32. What numbers
must be added to those in the scale in Art. 329 to enable you to play
these triads? How nearly can you produce these triads with the
tones of the tempered scale. Art. 331? Repeat the calculation for
the triad beginning on 6 = 45 and the one whose middle note is g =• 36.
How do these triads fit the tempered scale?

5. A string under a tension 0+-600 gms force, gives middle C (Ci).
Under what tension will it give J^i? Gi? C2?

6. A string 60 cm long and 0.5 mm in diameter gives Fj, what
must be the diameter of a string of the same length and under the same
tension in order that it may give A^? C2? A2?

7. A violin bow is drawn across the top of a narrow strip of spring
brass 10 cm long, and it gives a certain tone, say C2. Consider it as a
rod: to what length must it be reduced in order to give the octave C3?

8. A steel rod 80 cm long, clamped at the middle and rubbed with
a rosin cloth gives the tone A3. What changes in its length will cause

356 PHYSICS

it to give successively the other seven notes of the scale beginning on
this note?

9. An organ pipe is 8 feet long. What must be the length* of a
pipe, all other things being equal, that will give the fifth below the note
given by the first? The octave above?

SUGGESTIONS TO STUDENTS

1. If you have a banjo, a guitar, or a mandolin, measure the dis-
tances from the frets to the bridge and see if they satisfy the laws of
vibrating strings and the vibration numbers of the major scale. Can
you find out whether the frets are tuned to the tempered or the pure
scale?

2. If you play a violin, or have a friend who does so, play the har-
monics and get your friend to measure the distance of your finger
from the nut or bridge. Is this distance always an aliquot part of the
length of the string? Can you recognize the pitches of the har-
monics produced, using, if necessary, a piano to assist you? Do the
vibration numbers of these notes "check up" with the lengths of the
strings?

3. Can you find out, with the help of a standard tuning fork, whether
your piano is up to concert pitclj? Are all pianos and organs really
tuned to the same pitch? When your piano is being tuned, consult
the timer and find out if he uses beats to determine where the two
strings are in tune. Can you find out how organs are tuned?

4. Can you make a musical instrument that will play the scale,
by driving pieces of knitting needles into a board? If you succeed,
measure the lengths of these rods, and see if they follow the law for
transverse vibrations as stated in Art. 321. Examine a musical box
and see if it is made on this principle.

5. See if you can cut a long rod of dry, elastic wood into pieces of
such lengths that they will play the scale when you lay them across
two wedge-shaped sticks and strike them with a light hammer. Look at
a xylophone in a music store and see if this principle applies to it. Does
the tone depend on where the supports are placed ?

6. How are the vibrations of a violin string communicated to the
body? What part has the air in the body in producing and sustaining
the tone?

7. Try to get a loud sound from a wire stretched between two iron
gate posts. Does the result indicate that the air is set into vibration
by the string of a violin, or is it the body that does this work?

8. What can you find, by examining the sound-board of a piano, as
to the way in which it is adapted in shape and construction so as to
give resonance to notes of various pitches?

CHAPTER XVIII
HARMONY AND DISCORD

337. Wave Shape and Tone duality. We shall devote this
chapter to the discussion of the last point in our investigation into
the physical basis of harmony and discord. This point is in-
volved in the question, why do we call notes like c and /# discord-
ant when they produce, when sounded together, as many as 120
beats per second, while no discord results when the beats are less
than 4 or more than 30? In order to answer this question, we
must recall some of the facts presented in the preceding chapters.

First, waves bring us information as to: 1, the direction of the
source; 2, the number of vibrations of the source; 3, the intensity
of vibration of the source; and 4, the nature of the vibration of
the source. We have further identified these four kinds of in-
formation with the characteristics of waves, as follows: 1, the
direction of propagation depends on the direction of the source;
2, the length of the wave is connected with the number of vibra-
tions or pitch of the source; 3, the amplitude of the wave is derived
from the amplitude of the vibration of the source; and 4, the
shape of the wave varies with changes in the nature of the vibra-
tion of the source.

When we apply these facts to sound, we see that the direction
of propagation tells us of the direction of the source of sound,
that the wave length tells us of its pitch, and the amplitude tells
us of its intensity. But what information do we derive from
differences in the shape of the sound waves? How do we detect
these differences in shape? Since the only characteristic of
sound remaining for determination is its quality, and the only
characteristic of waves remaining undetermined is their shape,
we may conjecture that our perception of differences in qualities
in sound is dependent on our perception of differences in the
shape of the sound waves.

357

358

PHYSICS

Let us then adopt the hypothesis that tone quality is connected
with wave shape, and see whether it will help us in getting the
answer to the problem before us. The first step in the discussion
of this question is that of determining how differences in the
shapes of waves are produced. This we have already done, for
we have learned that waves of complex shape are produced by
adding together simple homogeneous waves of different lengths,
amplitudes and phases (cf. Art. 300). Hence we can conceive
that a sound wave of complex shape would result from the addi-
tion of two or more simple sounds differing from one another in
pitch and intensity.

Fia. 192. The Vibratinq

FLAM£

338. The Vibrating Flame. That this conception corresponds
with the facts may easily be shown by experiment. We have but
to devise a scheme for rendering the motion of the air particles

visible, and then to bring several sources
of simple homogeneous waves together,
to see if the resultant motion of the air
does not indicate that we have a com-
plex wave. ^Probably the simplest
method of doing this is th^ following:
A thin rubber membrane AB (Fig. 192)
is mounted between two rings of wood.
A flexible tube C leads the sound waves
up so that they can act on one side
of this membrane. The membrane will then follow the vibra-
tions of the air in the tube C. On the other side of the membrane
is a small gas chamber D. Illuminating gas flows into this cham-
ber at F and bums at the jet E. Whenever the membrane AB
vibrates, the gas in D will vibrate also; and this will cause the
flame to vibrate, the tip of the flame following roughly the vibra-
tions of the membrane. Since the vibrations of sound are too .
fast to be observed by the imaided eye, we have to observe
the flame in a nrirror which is kept in rotation. The apparatus
ready for use is shown in Fig. 193. When thus observed in the
rotating mirror and no sound is acting, the image of the small
fliame appears to be drawn out into a straight band of light; but

HARMONY AND DISCORD

359

when a train of sound waves is allowed to strike against the mem-
brane, this band is no longer straight, but its upper edge assumes
a wave-like form which must correspond closely in shape to that
of the waves impressed on the membrane.

Let us first send in sound waves from the tuning fork (Fig. 193),
which, as we have learned, produces waves that are nearly homo-
geneous; the appearance of the flame in the rotating mirror will
be then shown in the top band in Fig. 194. Using a second tuning

Fig. 193. Apparatus for Observino Vibrating Flames

fork, an octave below the first, the appearance of the* flame will be
that shown in the second band in the figure. If now we send in
the waves from both these tuning forks at the same time, the
appearance of the flame in the rotating mirror will not be the
same as before, for we have now added together two waves of
different periods, and therefore have a complex wave. The
result is shown in the third band in the figure. Referring to curve
R in Fig. 181, page 309, we see that the shape of the top of the
band of light agrees roughly with the shape of the curve there

360

PHYSICS

obtained as the resultant of two waves, one of which had half the
period of the other.

We thus prove that two or more simple homogeneous sound
waves add themselves together just as other waves do, and pro-
duce resultant waves of complex form.

339. Are Musical Tones Complex? The question then arises,
are the waves sent out by piano strings, violin strings, or the human
voice simple homogeneous waves, or do they have complex forms?

If they are complex, do two
complex waves of the same
period, but corresponding to
tones of different quality,
have different shapes? The
answer to this question is
easily obtained from obser-
vations with the little vibrat-
ing flame. For if one of us
sings into the flexible tube C
at the same pitch the vow-
els a, o, the appearance of
the flame in the rotating mir-
ror will be as shown in the
fourth and fifth bands in
Fig. 194. This result is a
most striking confirmation of
the hypothesis that tones of
the same pitch but of differ-
ent qualities have wave forms of different shape.

The tones from strings, organ pipes, and other musical instru-
ments, when analyzed in this way with the vibrating flame, show
differences in wave form corresponding to their different qualities.
But waves of different shape are produced by compounding simple
waves in various ways. Therefore we see that a complex tone
must be produced by the addition in various ways of simple tones,
and that the quality of the complex tone depends on the way in
which the various simple tones happen to be brought together.

^^UUUUitM

{MMMMitdM^

(fM^/fM/4Aj(4M/44jL

f^Lf/iLMM^-^^LfdHi

Fig. 194. Appearance of the Vibrat-
ing Flame

HARMONY AND DISCORD 361

340. How Musical Tones axe Possible. It is easy to see how
complex waves may be produced by the addition of two or more
simple homogeneous waves of different lengths which originate
from different sources, as from two or more tuning forks. But
we have just learned that a single vibrating body, like the human
vocal organ or a musical instrument, produces such complex
waves. How can a single vibrating body produce several different
vibrations at the same time? And if it does do so, are there any
relations among the different vibrations which are thus produced
at the same time? Recall the jumping rope (Art. 301). We
learned that it may vibrate in one loop, in two loops, in three loops,
depending on how rapidly it is turned. Similarly, a stretched
string (Fig. 195) may vibrate in one loop, in two loops, in three

Fig. 195. The String May Vibrate in Three Loops

loops, etc. Since in this case the string is stretched with a con-
stant force, and since the lengths of these loops are 1, i, J
the length of the string, etc., the vibration numbers of the notes pro-
duced are related by the simple ratios, 1: 2: 3, etc. Suppose
it were to vibrate in several of these ways at once, what would
be its shape? We can find out by adding together the com-
ponent vibrations as in Art. 300. Thus, if we conceive the string
to be vibrating in one and in two loops at the same time, and that
the amplitude of the 2 loops is only half that of the 1, the result
is shown at R in Fig. 181. If now, in addition, it is vibrating in
three loops, and the amplitude of the 3 is but J that of the 1, we
add the 3 loop wave to the resultant of the other two and obtain
the curve shown at iJ' in Fig. 181. Similarly, by adding the 4
loop vibration with J the amplitude of the 1, and the 5 loop vi-
bration with ^ the amplitude of the 1, we get the resultant shown
at P and Q, Fig. 196. If we continue this process of adding the
curves that correspond to greater numbers of vibrations, each

362

PHYSICS

with a correspondingly smaller amplitude, the resultant becomes
more and more like the curve at R in the figure.

We thus see that a string may vibrate in all these ways at
once if it can take the shape shown at R. But can strings take
that shape? What would be the shape of a stretched string if a

piano hammer had hit it
at a point near the end?
How does the violin bow
act? Does it not pull the
string into a shape similar
to that shown in the figure
until the tension of the
string becomes great
enough to overcome the
friction of the bow? Then
the string flies back. Or, if we merely pick the string with a sharp
point, as in the case of the mandolin, we bring the string to the in-
dicated shape and then let it go. So we see that a string may be
made to take the indicated shape, and therefore we may infer that
it can send out a compound wave similar to that composed of a
number of vibrations whose periods are related to one another,
as 1: 2: 3, etc., and whose amplitudes continually decrease.

Fig. 196. Complex Waves op a String

341. Fundamental and Overtones. We must now distin-
guish between the tones thus produced. For this purpose we
call the tone that corresponds to the vibration of the string in one
loop the FUNDAMENTAL. It has the smallest number of vibrations
and is most intense of them all. The other tones are called over-
tones, or harmonics. Since the number of vibrations of the string
in two loops is twice that of the string in one loop, the first over-
tone will be the octave of the fundamental. Similarly, the second
overtone is related to the first overtone by the ratio |, and will
therefore be to the first as ^ to c in the musical scale. This
interval is called a fifth. Similarly, the third overtone is related
to the first overtone by the ratio |, and will therefore be an octave
above the first, etc.

There are several interesting things about these overtones.

HARMONY AND DISCORD S6S

In the first place, the quality of a complex tone evidently depends
on which overtones are present and how strong each is; for we
have shown that notes of different quality produce complex waves
of different shapes, and also that differences in shapes of waves
are produced by differences in the number and strength of the
simple waves of which they are composed. In the second place,
we can show that a simple wave produces resonance in a body
whose natural period agrees \iith that of the simple wave, not
only when the simple wave exists alone, but also when it is a
component of the complex one.

342. Overtones of Piano Strings. The simplest way of showing
this is the following: Press a key, say middle c of the piano,
gently, so that the hammer does not strike the string, but so that
the muffler is lifted. The string will then be free to vibrate by
resonance. Then strike the key C, an octave below, and let it
rise again so that the muffler stops the vibrations. The tone
middle c will be heard gently humming in the piano. But we
have just learned that the lower C contains the upper c as its
first overtone; so we see that, since the overtone c exists as part
of the compound tone C, the string whose natural period agrees
with that of this overtone is set into vibration by resonance. Simi-
larly, if we prcss the note g above middle c so as to lift its muffler,
and then again strike the lower C, the tone g will be heard coming
from the piano, showing that g exists as an overtone in the vibra-
tions of C If, however, the experiment be tried with the note /
above middle c, no tone will be heard from the piano after lower
C has ceased to vibrate, because / is not an overtone of C, and
therefore the vibrations corresponding to / do not exist in the
complex tone emitted by C

343. Helmholtz Eesonators. The experiment is even more
striking if we use as resonators, not the strings of the piano, but
a series of hollow brass spheres, whose volumes are such that the
natural periods of the volumes of air in them coincide with the
periods of different notes of the piano. Such a series of hollow
spheres was used by Helmholtz in analyzing these complex tones.

364 PHYSICS

One of them is shown in Fig. 197. The little projection on one
side is intended to be fitted into the ear, thus enabling it to detect
very faint sounds whose periods agree with that of the resonator.

344. How the Ear Perceives a Complex Tone. Let us now

expand our conception of the resonance effect of a compound

tone to include all the overtones at once. What effect will be

produced on a series of strings, tuned to all the

'^^ notes of the scale and free to vibrate, if we

i. ^^k sound near them a compound note whose f un-

m^* Jj^^^^ damental agrees with one of the notes of the

^Bij^^^v scale? Evidently those strings that correspond

^^^^^ to the overtones will be set into vibration by

Fig. 197. Reso- resonance, while the others will remain at rest.

NATOR 1 1 rt. 1.

Thus we see that the etiect of a compound tone
on such a series of strings, like those of a piano or harp, is similar
to that produced by a skilled hand passing rapidly over the strings
and touching gently those among them that correspond to the
overtones and the fundamental note.

We have learned that the ear contains such a series of strings
or fibers, and so we may imagine that when a complex note falls
on the ear, not all of these fibers are excited, but only those whose
natural periods agree approximately with the periods of the fun-
damental and the overtones of the note.

We may now reach our final conclusion concerning discords;
for since it appears that all musical notes are complex, and since
5uch a note excites in the ear not only the fibers corresponding
nearly to its fundamental, but also those corresponding to the
overtones, it becomes clear that to obtain harmony between two
notes we must avoid beats, not only between the fundamentals,
but also between the overtones. Therefore the complete answer
to our question as to the reasons for discord is, two tones are dis-
cordant when either their fundamentals or any of their overtones
produce beats which are more than 4 or less than 30 per second.

It now remains for us to show that this principle will enable
us to make clear why the interval c-g is more pleasing than c-/#.
In order to do this we have merely to write out the numbers of

HARMONY AND DISCORD 365

Vibrations of the fundamentals and of the overtones and see
whether we have such beats anywhere. These numbers are, for
the three notes under consideration,

256

512

768

1024

1280

1536

384

768

1152

1536

376

752

1128

1504

It thus appears that the discord between c and /# is due to the
production of 16 beats by the second overtone of c and the first
of/#.

345. Belated Tones. This table brings to light another in-
teresting fact concerning the notes c and ^r, viz., that some over-
tones are common to both. We see that both have an overtone
of 768 vibrations and another of 1536. Noting this fact, Helm-
holtz calls such notes musical relations, i.e., he says that when two
tones have two or more overtones in common, they are musically
related. Such musical relationship must occur between notes
whose fundamental vibration numbers are related by the simple
ratios 1: 2: 3: 4: 5: 6, etc., because the vibration numbers of the
overtones are related to those of the fundamentals by these same
ratios.

And so, at last, we reach the answer to Pythagoras's problem.
It may be stated in many ways, but perhaps the simplest is the
following: The numbers of vibrations of the overtones of strings
and air columns are related to those of the fundamental by the sim-
ple ratios 1: 2: 3, etc. Therefore the notes of the scale must be
related by the same ratios in order to avoid disagreeable beats be'
tween both fundamentals and overtones.

346. Chimes. Do we ever use other sources of musical tone
besides strings and air columns? We might answer, ''Yes," and
cite, as an example, chimes of bells, which are justly reputed to
produce a decidedly musical effect. But did you ever hear a chim^i
of bells played in chords, i.e., more than one note at a time? Prob
ably not, because the overtones of the bells are not related to th^ir
fundamentals by the simple ratios 1: 2: 3, etc., and so when bells

866 PHYSICS

are played in chords, the effect is musically intolerable, because
disagreeable beats occur between the overtones.

Another interesting conclusion is, that fundamental tones
which have no overtones would not be disagreeable, when the same
fundamental tones with overtones would be so. This is easily
shown to be true by sounding together two tuning forks with
pitches c and /#, for instance, and comparing the effect with that
produced by two organ pipes or strings of the same pitches. The
combined effect of the forks is not at all disagreeable, while that
of the pipes or strings is decidedly so.

There are many other interesting and perplexing questions
concerning tone quality and concerning harmony and discord.
For example, how can we control the tone quality, as in the organ,
where we make pipes whose tones resemble flutes, violins, horns,
and even the human voice? How are the wonderfully different
qualities of the human voice produced? If we could magnify
the records cut by a phonograph in the wax cylinder, what would
their shapes be? How are the possible successions of chords
in a musical composition dependent on the tone quality and the
beats. These inquiries can' not be pursued here, for a dis-
cussion of them would lead us far beyond the scope of this book.

SUMMARY

1. Tone quality is related to wave shape.

2. The addition of simple sound waves produces waves of
complex shape.

3. A single vibrating body may send out complex waves.

4. The complex vibrations of strings and air columns are
composed of simple vibrations whose numbers are related by the
ratios 1: 2: 3, etc.

5. The lowest note in the complex tone is called the funda-
mental and the others are overtones.

6. The relations between the vibration numbers of the
fundamental and the overtones of strings and air columns are
expressed by the ratios 1: 2: 3.

7. Tone quality depends on the number, pitches, and rela-
tive intensities of the overtones.

HARMONY AND DISCORD * 367

8. An overtone in a compound note may produce resonance
just as if it were alone.

9. Two tones are discordant when either the fundamental or
any of the overtones combine to produce disagreeable beats.

10. The notes of the musical scale must be related by simple
ratios because the overtones of musical instruments are so related.

11. Harmony depends on tone quality as well as on pitch.

QUESTIONS

1. Why may we assume that tone quality and wave shape are
related?

2. How do we prove that this assumption is correct?

3. How can a single vibrating body send out complex waves?

4. How do we know that the numbers of vibrations of the over-
tones of a string are related by the simple ratios 1: 2: 3, etc.?

5. Upon what does the tone quality depend?

6. Can one component in a complex wave act to produce resonance
in a body whose number of vibrations agrees with its own?

7. When the fundamentals do not produce disagreeable beats,
why may two tones still be discordant?

8. What do we mean when we say two tones are musically related?

9. Why do the notes of the musical scale have to be related by
simple ratios because the overtones of strings and air columns are so?

10m Could we replace" the strings of a piano with bells with good
musical effect? If not, why not?

PROBLEMS

1. Beginning with c = 24, write out the vibration numbers of
the first 8 overtones of a string of that pitch. Do the same with the
tone g = 36. How many of the eight overtones are common to both
tones? Which of the overtones is the first common one? Write out
the first eight overtones beginning on / = 32, and also on e = 30. How
many overtones has each of these tones in common with the series
beginning on 24? Which of the overtones in each series is the first
common overtone? Which is the best consonance, c-g, or c-e? Which
pair have the greatest number of common overtones?,

2. Have the two tones beginning respectively on c = 24 and d =
27 any of their first eight overtones in common? Do any of their over-
tones give disagreeable beats, i.e!, more tlian four and less than 30 per
sec? Is this interval c-d more or less consonant than the interval c-g?
Can you see any connection between the consonance of a musical interval

368 PHYSICS

and the number of the first overtone wliicli is common to the two
component notes?

3. In a string on a musical instrument a node cannot exist at the
point where the string is either bowed, picked, or struck with a ham-
mer. If a string is plucked at a point distant from the bridge J the
length of the string, what overtones will be wanting in the tone pro-
duced?

4. A piano hammer strikes at a distance of } the length of the
string from one of its ends. What overtones are wanting in the tone
produced? Write the series of overtones for c = 256, and see if the
seventh is apt to cause beats with its neighbors. Can the quality
of the tone of a piano string be varied by changing the position of
point where the hammer strikes? Why does a violinist bow near the
bridge when he wishes to produce "brilliant" tones?

SUGGESTIONS TO STUDENTS

1. Construct a vibrating flame as described in Art. 338, in Hop-
kins's Experimental Science j and in Mayer's Soundj and see what vowel
sound gives the most interesting vibrations. See if each of your voices
gives the same shaped flame when singing the same vowel on the
same pitch. Try other musical instruments in the same way. See
if you can photograph the flame.

2. Can you find out how a phonograph or a graphophone works?
What sort of curves must be cut in the cylinder or disc of the machine?
Have you ever examined such a curve with a microscope?

3. Ask the organist at your church how organ pipes are made to
have different qualities of tone. Examine the pipes yourself and see
if you can think why the diameters of the flute and violin pipes are
smaller in proportion to their lengths than those of the diapason pipes.

4. Pronounce the vowels. In which is the mouth cavity elongated
so as to give resonance to the lower overtones? In which is it short-
ened so as to cut them out?

CHAPTER XIX
LIGHT

347. What Does Light Do for TJs ? A peculiar interest at-
taches to the study of light, because of its great usefulness to
mankind. Not only is it indispensable for all human action,
but also the color combinations by which it enables us to express
art ideals are sources of highest pleasure and satisfaction. Can
you conceive of a world devoid of light? And what a monotonous
existence we should lead if light were deprived of color I Yet
the very omnipresence of light often leads us to overlook its vast
importance to life in the universe. In taking up the discussion
of this subject, then, let us ask, first, what does light do for us?

When we ponder this question carefully, we are led to con-
clude that light enables us to gain information of four different
kinds. 'First, it enables us to distinguish differences in the
directions in which objects are located with reference to us and
to one another. This ability to recognize differences in direction
enables us to determine the shapes of objects as well as their
relative positions, for the different parts of an extended object
lie in different directions from our eyes.

In the second place, light makes it possible to distinguish
between the colors of things. This power not only assists us in
distinguishing between objects about us, but it also enables us,
as we shall presently see, to observe the peculiarities of distant
stars and study the mechanism of ultimate atoms.

In the third place, we are able to distinguish between intense
and faint light — to recognize all the possible gradations of light
and shade, whose totality produces the pictures which succeed
one another with endless variety and which produce in us emo-
tions of joy or pain, of inspiration or dejection, throughout
our entire conscious lives.

And, lastly, we can not only appreciate simple color, but we can

309

370 PHYSICS

distinguish and produce endless shades and varieties of color by
mixing the simpler colors together in different ways. This last
power, which we could not have without light, is fundamental in
the art of painting, and is thus of far-reaching importance in
our appreciation of the beautiful both in nature and in art.

Now, it may seem to many sacreligious to attempt to pry into
the mechanism of light — to seek to find out how light is able to
do all this for us. We must confess that we think that it would
be so if it were not for the fact that this inquiry does not in any
way destroy our recognition of the enormous utility of light nor
diminish our appreciation of the beauties of nature and of art
which it enables us to enjoy. On the contrary, the detailed
study of the phenomena of light, in the way in which the physicist
studies it, adds enormously to our estimation of the wonders of
light by showing us the ingenious way in which it operates to
serve us as it does.

348. What is the Nature of Light? After noting carefully
the common experiences with light, the first t*hing the physicist
does, is to ask what assumption or hypothesis he can adopt that
will enable him to group the various phenomena together and to
construct a mechanical model that will assist him in describing
its action more in detail. When asked to propose such a hy-
pothesis, what shall we say? We have just analyzed the kinds
of information that light helps us in acquiring, and find them
to be of four sorts, viz.: 1, As to direction; 2, as to color; 3, as
to intensity; and 4, as to blending of colors. What sort of mech-
anism is able to bring us such information? Probably, just as
in the case of sound, a wave motion would suffice; and therefore
we will at the outset adopt the hypothesis that light is a wave
motion.

But what characteristics of the phenomena of light may we
identify with each of the characteristics of the wave? Cleariy,
the sense of direction is derived from the direction of propagation
of the waves. It is also clear that the inttmsity of the light cor-
responds to the amplitude of the waves. This leaves perception
of color and the composition of colors to correspond respectively

LIGHT 371

to the wave length and the wave form. Possibly simple color
may correspond to wave length and complex color to wave
form. Let us, then, assume that these are the relations and see
to what conclusions we shall be led. To this end we must enter
upon a more detailed discussion of these characteristics of light.

349. Direction. The first question that naturally arises
is, how do we detect differences in direction? You answer,
"With our eyes"; but how do they operate to detect the direction
of propagation of waves? In order to answer this question, we
must first call to mind a very familiar characteristic of light, viz.,
that it appears to travel in straight lines. Thus, the sunlight
falling on the floor traces an outline of the window there. If we
cover the window with a shutter having a small hole in it, we.
notice that the beam of light which passes through the hole, and
whose path is revealed by the dust particles in the air, travels
in a straight line, and makes a bright spot on the floor. We also
notice that the path of the light is the continuation of the line
joining the sun and the hole in the shutter, so that if we invert
the process and draw a line from the spot on the floor to the hole,
that line indicates the direction of the sun. Thus we see that
we can determine the direction of the sun with reference to the
shutter and the floor, because the light travels ordinarily in a
straight line.

350. Image. If now we have two bright objects outside the
window, like two electric lights, each will produce a bright spot
on the floor or on some suitable screen held behind the hole in
the shutter. When we draw straight lines from these two spots
to the hole, they inclose an angle between them, and by means
of this angle we are able to judge of the relative positions of
the two electric lights. If now we have a large number of such
bright points, for example a landscape outside the window, each
point of the landscape produces on the screen a bright spot, which
indicates the direction of the point with reference to the screen
and the hole; these bright spots on the screen will each indicate
the direction of its corresponding source, and so we obtain on the
screen an image of the landscape outside (Fig. 198).

372

PHYSICS

Two things are apparent concerning the image thus formed:
1. It is inverted, and 2, it is indistinct* and fuzzy. A moment s

FiQ. 198. The Image is Inverted and Fuzzy

thought will show us why it is inverted, namely, because the rays
all cross at the hole, so those that were below on one side are now
above on the other, and vice versa. Therefore this characteristic
of the image is inherent in the nature of the phenomenon and
can not be altered.

But why is the image fu^zy? An analysis will show us why.
It is because each point of the land-
scape is sending out waves which
spread out in all directions about that
point (c/. Fig. 172). When these waves
reach the hole in the shutter, they are
divergent, and therefore make on the
screen a spot of light somewhat larger
than the hole, as shown in Fig. 199.
Thus the image of each point of the
object is a spot of light on the screeen,
not a point; and therefore the entire
image, which is the sum of these
spots, is not clear and distinct like the object. Yet, even so, the
image is enough like the object to be readily recognizable.

Fio. 199. The Spot is Larger

THAN THE HoLE

LIGHT 373

351. What the Eye Does. Now, although this image is not
as clear as the landscape outside, it enables us to distinguish
definitely the directions of the different points of the latter. We
might, therefore, conceive that the eye is able to distinguish
directions by a similar device; for is not the pupil of the eye
merely a small hole in a shutter, and therefore there must be
formed at the back of the eye an image of the object in front?
Now, an examination of the construction of the eye shows that
immediately behind the pupil there is a little transparent object
of hard elastic substance, called the crystalline lens (L, Fig.
200). The front and back of this lens seem to be portions of

Fig. 200. The Image Formed in the Eye

spherical surfaces, and it is thicker at the middle than at the
edges. Behind this lens the eye is filled with water, and the rear
surface, called the retina, is covered with fine nerve filaments.

352. What a Lens Does. We can see that an image would
be formed at the back of the eye without the crystalline lens.
What, then, is the use of this addition? On holding a piece of
glass that is shaped like the lens of the eye behind the hole in the
shutter, we observe that when the screen is at one particular
distance from' the hole the image of the landscape outside is very
clear and distinct. Therefore we may conclude that the purpose
of the crystalline lens in the eye is to render the image on the
retina distinct, i.e., to bring the rays from a point on the object
outside together on the retina in a point instead of in a spot. Re-
ferring to Fig. 204, p. 377, we see that such a lens must be able to

374

PHYSICS

bend the light, so that, after passing the lens, it is convergent
instead of parallel or divergent.

Fig. 201. The Light is Bent
WHEN IT Enters the Water

353. How Light is Changed in Direction. But if light
travels in a straight line, how can a lens bend it? Yet, clearly, it

does do so. Have you ever noticed
that Ught bends when it passes
obliquely from air into water? Place
a battery jar full of water so that
the sunbeam from the hole in the
shutter falls obliquely on the surface,
Fig. 201. Does the path of the light
have the same direction in the water
as it does in the air? Is [the path
in the water straight? Thus it be-
comes clear that light travels in a
straight line only so long as it is
moving through the same sort of matter, for when it passes from
air to water it is bent. A similar effect is observed when we
pass the light into glass or into any other transparent substance.
How may we conceive that this bending is effected? We have
assiumed that light is a wave
motion. Let us then imagine
that we have a beam of light
of width a b, Fig. 202, traveling
in air and approaching a sur-
face of water a c. Let the direc-
tion in which the light is trav-
eling be represented by b c.
Then the front of the wave, i.e.,
the line joining those points of
the wave that are in the same
phase, will be represented by

a by which is perpendicular to b c. When the light has entered the
water, we find that it is traveling in the direction c e; so that the
wave front, which in the water is perpendicular io ce the new direc-
tion of travel, has been turned from the direction a 6 to that of c d.

Fig. 202. Refraction

LIGHT 375

This result would be accomplished if that portion of the wave near b
traveled the distance 6 c in air in the same time that was taken by
the portion of the wave near a to travel the distance a d in water.
But ad \s clearly less than b c. So we see that we can form a
perfectly intelligible picture of the manner in which a ray is bent
on passing from one medium to another, by assuming that the
light waves travel more slowly in the water than in the air.

354. Index of Eefrax^tion. This phenomenon of the bending
of a beam of light when it passes from one mediium to another
is called refraction. How can we measure it? Clearly, the
amount of bending depends on how much difference there is be-
tween the velocity of light in the two media. For if 6 c remains
the same, then the less ad is the greater the bending will be.
We may, therefore, measure the bending by the ratio of 6 c to
a d. But a d represents the distance traveled by the light in the
second medium in a certain time t, and b c represents the distance
traveled in the first medium in the same time t Therefore ad
and b c are proportional to the velocities of light in the two media.
So we may say that the amount of bending depends on the
ratio of the velocities of light in the two media. Since we can
measure the amount of bending by the ratio of these two veloci-
ties, we call that ratio the index of refraction. It is usually
denoted by n. Therefore we define this index in the following
way:

_ velocity in first medium _ b c ,

velocity in second medium ad'

It has been proved by experiment that the velocity of light in a
given medium is constant for a given color, therefore we may
infer that this index remains constant for the same two media.
It is not always easy, however, to measure the velocities in
the two media; therefore let us see if there is not a more
convenient form of expressing this ratio. To do this, drop a
perpendicular n(m/ to the surface between the two media (Fig.
203). Then the angle neb, formed between this perpendicular
and the direction in which the light is traveling in air, is called

370

PHYSICS

the ANGLE OF INCIDENCE and is usually denoted by i. Similarly,
the angle nfce is called the angle of refraction and is usually
denoted by r. Now, from the figure we see that ^ neb = ^ bac = i
and ^nfce = ^acd = r. We have also learned (c/. Art. 304) that

itin bac = — = 9in i, and sin acd = — = sin r, whence
ac ac

sin I
sinr

be
ae

sin I

^^ = ^ = ^. Therefore [ef. equation (18)] n^ ^^ (19).
sin r ad ad ^ sinr ^

Thus tlw index of refraction is -equal to the sine of the angle of
incidence divided by the sine of the angle of refraction. Since

these angles are easily measured, this
ratio is more convenient to use than
that of the velocities.

Now, if we use light of one color
and measure the angles r that cor-
respond to various angles of incidence
i, and then, with the help of a table
of sines, find the corresponding values
of 71, we shall find that the values of
Fig. 203. Refraction Diagram »i are the same for all angles of inci-
dence. Therefore we conclude that
n is constant for any two media and for any definite color ^ as
we have inferred that it should be. . This fact was discovered
experimentally by Willibrode Snell in 1680, and is known as
snell's law, or the law of refraction.

356. How the Lens Forms the Image. We are now able
to see why the introduction of a lens of the form of the crystalline
lens of the eye improves our image; for since that lens is thicker
in the middle than it is at the edges, it is evident that those por-
tions of the wave which pass through the center of the lens travel
through a greater thickness of glass. But the light travels more
slowly in glass than in air, so the center of the wave is retarded
more than the portions near the edges of the hole, and the
wave is converted from a plane or a convex wave into a concave

LIGHT

377

■> ^ ■>

5 -o

FiQ. 204. The Light is Brought to
A Point

wave, as shown in Fig. 204, in which the vertical lines represent
the successive wave fronts.

Because . after passing the lens the wave fronts are concave,
they contract toward a point 0, and there form a small image
of the point from which they
started. Thiis we see that
introducing a lens of the
given form contracts the
spot of light to a point.
Therefore every luminous
point of the landscape is
represented by a single

bright point on the screen, and so the image becomes brilliant
and distinct.

We see, however, that the screen must be placed at the definite
distance from the lens in order to receive a distinct image.
This distance from the lens to is called the focal length, and
the point is called the focus. Manifestly, the focal length will
depend on the curvature of the lens, its index of refraction, and
the shape of the incident wave. The study of the relations be-
tween these quantities is of great importance, for the construction
of all optical instruments depends on them. We can not properly
understand optical instruments without a clear conception of these
relations. We shall take up this study in the next chapter, as

our attention is first demanded by one
other important phenomenon con-
nected with our perception of the
direction of light.

Fig. 205. Reflection

356. Reflection. We can best
study this phenomenon by placing a
mirror on the floor where our sun-
beam falls. The beam is turned^
away from the floor and reflected to the ceiling or to some
other part of the room. Fig. 205. If we turn the mirror into
different positions, we note that the reflected beam is turned
in different directions. But by so turning the mirror we vary

378

PHYSICS

the angle at which the incident beam falls upon it. We also
vary the angle at which the reflected beam leaves it. Is there
any relation between these two angles?

!Fi6. 206. Reflection Diagram

367. Laws of Eeflection. Before answering this question, we
must agree as to how we shall measure the angle of incidence. We

therefore define the angle of inci-
dence to be the angle included be-
tween the perpendicular to the sur-
face and the incident beam. This
angle is measured in the plane
which contains this perpendicular
and the incident beam. Thus if
(Fig. 206) AB represents the direc-
tion of the incident beam, BC that
of the reflected beam, and NB the perpendicular to the mirror, the
angle of incidence is then NBA, This angle is, as before, denoted
by i. Similarly, the angle of reflection is that included between the
perpendicular NB and the reflected beam 5C, i.e., it is NBC.

We may now ask what relation, if any, exists between these
angles. In order to answer the ques-
tion, we must measure various angles
of incidence and the corresponding
angles of reflection. If w^e do this,
we find that tJie angle of incidence is
equal to the angle of reflection in all
cases. We further find that the in-
cident beam, the perpendicular, and
the reflected beam, all lie in tJie same
plane. Therefore we conclude that these are the laws of re-
flection.

Fig. 207. The Plane Mirror

358. Where the Image Appears. These principles can now
be used to find out where a source of light aj)pears to be when we
observe it by reflection in a mirror. Clearly, an observer at C,
Fig. 206, will receive the light as if the source were in the direction
CB, But at what point in that direction? In order to find out,

LIGHT 379

we have but to alter Fig. 206 as follows: Let S (Fig. 207) be the
source of light. Since it is sending beams in all directions, it
will send out not only one in the direction S B, but also many
in other directions. One particular ray SD will strike the
mirror perpendicularly, and be reflected back along its own path,
i.e., in the direction of DS.

Now, an observer at C sees the light in the direction CB and
another observer behind S and in the direction DS sees it in the
direction SD. In what direction does the source appear to lie?
Clearly in the directions of both lines. Hence the apparent source
of the light must be at the point S', where the lines CB and SD,
intersect when they are extended behind the mirror.

But where is the point S' with respect to the mirror? From
the law of reflection, we know that the angle NBS == angle NBC,
hence we may easily prove that the two right triangles SBD and
S^BD are equal, and therefore S'D = SD, But these are the
distances of the source S and its image S', measured perpendicu-
larly from the mirror MM; so we see that this result may be stated
as follows: When light is reflected in a plane mirror the image
appears to he as far behind the mirror as the source is in front of it.

359. Diffuse Eeflection. The law of reflection just stated
applies clearly to all cases of reflection from metallic or other
polished surfaces. If we replace the mirror by a piece of white
paper, what becomes of the reflected beam? If we try the ex-
periment with other things by placing them in the path of the
sunbeam we will notice that some of them act like the mirror and
reflect most of the light in a definite beam while others reflect
part of it in a definite beam, and still others reflect none of it in a
definite beam. Hence we see that this law of reflection is not
general in its application. We may, then, ask what law applies
to these other cases. On placing a white card in the path of the
beam, we note that the light seems to be scattered in all directions
from the bright spot on the card. In fact, the effect is the same
as if the card were itself the source of the light. This phenom-
enon is called diffuse reflection in contradistinction to the
other kind, which is called metallic reflection.

380 PHYSICS

The importance of diffuse reflection is seldom appreciated.
We do not often realize that we see most objects because they
reflect diffusely. Thus, when we look at a landscape or a picture,
each part of the object affects us as if it were itself a source of
light. This, then, is the law of diffuse reflection, viz.: A body
that reflects light diffusely appears as if it were self-luminous.

We thus see that there are two kinds of reflection, metallic
and diffuse, of which the latter is the more important to mankind.
It is not possible, however, to classify all substances as reflecting
either metallically or diffusely. At one end of the series we have
the metals, which reflect in the first way only; and at the other
end we have what are called perfectly matt surfaces, like a plas-
ter wall, which reflect entirely diffusely. Between these two extremes
we have substances that reflect partly in one way, partly in the
other, in all sorts of varying proportions.

SUMMARY

1. Light enables us to acquire four kinds of information:
1. As to direction of a source; 2, as to its color; 3, as to its in-
tensity, and 4, as to its tone of color.

2. Waves also bring us four similar kinds of information, and
therefore we adopt the hypothesis that light is a wave motion.

3. Light ordinarily travels in straight lines in any one medium.

4. An image of an object is formed when the light from it
passes through a small hole.

5. Such an image is inverted and blurred.

6. A lens makes this image distinct and more brilliant.

7. The focal length of a lens is the distance from the lens to
the point at which the image is formed.

8. The conditions necessary for the formation of a clear image
are realized in the human eye.

9. The direction in which light travels is altered when it
passes obliquely from one medium to another.

10. The amount of this bending is measured by the index of
refraction.

IL The index of refraction is the ratio of the velocities of

LIGHT 381

light in the two media, or the ratio of the sines of the angles of
incidence and refraction.

12. The index of refraction is a constant for any two given
media and for a given color.

13. When light is reflected from a metallic surface, the angle
of incidence is equal to the angle of reflection, and lies in the
same plane.

14. The image of an object reflected in a plane mirror appears
as far behind the mirror as the object is in front of it.

15. Light is diffusely reflected from unpolished surfaces.

16. A surface reflecting diffusely appears as if it were itself
a source of light.

17. The reflection of light by many surfaces is partly diffuse
and partly metallic.

QUESTIONS

1. What four kinds of information does light enable us to acquire?

2. What may we assume as a working liypothesis as to the nature
of light?

3. Upon what property of light does our determination of the
direction of light depend?

4. How is an image formed through a small hole?

5. Why does a lens improve the clearness of such an image?

6. Describe the construction of the human eye. Wliat provision
is there made for distinguishing differences in the directions of objects?

7. Why is the path of light bent when it passes obliquely from air
into water?

8. How do we measure the amount of this bending?

9. What relation exists between the index of refraction and the
velocities of light in the two media? between that index and the angles
of incidence and refraction?

10. What is the focal length of a lens?

LI. What is the difference between metallic and diffuse reflection?

12. What is the law of diffuse reflection?

13. Which kind of reflection is the more common? Which is the
more aseful?

14. Where does an object reflected in a plane mirror appear to be?
Can you prove it by the geometrical relations?

15. What sorts of substances reflect metallically? What sorts
entirely diffusely?

382 PHYSICS

PROBLEMS

1. The method of finding the location of the image of a point source
in a plane mirror is described in Art. 358. Replace the point source
by an arrow and graphically construct the image. How far behind
the mirror does the image lie?

2. If Vi represents the velocity of light in glass and V2 that in water,
show that the index of refraction of light at a surface between glass

and water is equal to — . If v represents the velocity in air, what is

the index at the surface between air and glass? Between air and
water? May the index for glass-water be obtained by dividing that
for air-glass by that for air- water? Show how.

3. The index of refraction is shown to be the ratio of the sine
of the angle of incidence to the sine of the angle of refraction, Art. 354.
If the light falls perpendicularly on a surface of glass, so that the angle
of incidence is 0, what is the value of the angle of refraction? Is the
light bent when it passes perpendicularly through the dividing sur-
face?

4. Is the period of vibration of a light wave (i.e., the color of the
light) changed by passing from one medium to another, as from air to
water? With the help of the equation v = n Z, show that the index of
refraction n^ay also be defined as the ratio of the wave length of the
light in air to its wave length in water.

5. When light passes from water into air, is the path of the light
bent toward or away from the perpendicular to the surface? How
do you define the index of refraction under these conditions if Vj rep-
resent the velocity of light. in water and v its velocity in air? How
is it defined in terms of the sines of the angles of incidence and refrac-
tion? How does its numerical value compare with that for the con-
verse case of light passing from air into water?

6. The index of refraction of air- water has the value 1.33, i.e.,

— = 1.33, or sine i = 1.33 sine r, But'the sine of an angle is the

sxner

ratio in a right triangle of the side opposite the angle to the hypoth-

enuse, and since that side cannot be greater than the hypothenuse,

sine i cannot be greater than 1. What is the greatest value that sine r

can have?

7. If we reverse the direction of the light and send it from water
into air, is there any reason why the angle r should not be greater
than the value determined as a maximum in problem 7? If we give
the beam a greater inclination to the surface, so that the value of 1.33
sine r becomes greater than 1 , what becomes of sine i? What does
the light do? Can it escape from the water? This phenomenon is
called total reflection.

LIGHT 383

SUGGESTIONS TO STUDENTS

1. Using the principle that the angle of incidence is equal to that
of reflection, see if you can find out by graphical construction what
will become of a beam of parallel light after it is reflected from a con-
cave -spherical mirror. If you have a spectacle, camera, or opera glass
lens that has a concave surface, reflect the sunlight from it and see
if your construction is correct. Try the same experiment with a silver
spoon or a lamp reflector. Can you construct graphically in the same
way the image of an arrow as formed by a concave mirror?

2. Make a pin hole camera and see if you can take a picture with it.
Any light-tight box will do. To get fairly clear images, the edges of
the pin hole must be smooth.

3. Reflect a sunbeam into a neighboring window with a plane
mirror. When you turn the mirror through any given angle, through
what angle is the reflected beam turned? See if you can devise a method
of measuring it with a protractor and of making a graphical solution.
The geometry of the right triangle will aid you here.

4. How are search lights constructed? What is the shape of the
mirror?

5. Were you ever in a "crystal maze?" If so, explain your per-
plexities in a brief paper.

6. Why is a large mirror of advantage in decorating a small room?
Would a living room be comfortable if its walls were nearly covered
with metallically reflecting substances? Can you see a physical reason
why matt surfaces are considered in better taste?

7. Make a diagram to prove that the lower half of a full length
mirror is not necessary in order that a lady may see her entire figure
in it. Verify the conclusion by experiment.

8. Find out why most books are printed on matt paper.

CHAPTER XX

OPTICAL INSTRUMENTS

360. Principal Focus. In the last chapter we learned that
an image of a luminous point is formed by a lens at a particular
distance from the lens. Is this distance always the same for a
given lens, no matter where the luminous point is situated with
reference to it? In order to answer this question, take a simple
lens L of the shape shown in Fig. 208, and allow light from the sun
to fall on it in the manner there shown. On holding a paper
behind the lens, we easily find the point at which the image of
the sun is distinctly formed. Now, the
sun is so far away that the wave fronts
of the waves that reach us are sensibly
" plane. Hence they may be represented by
straight lines as AB, which are moving in
a direction LF perpendicular to AB. Since
BD AB \s a. straight line, all the perpendicu-

FiG. 208. Principal lars to it, which indicate the directions of
motion of the various parts of the wave, are
parallel to one another and to LF. These lines are called rays.
Such a series of parallel rays constitute a parallel beam. In
order to make the figure symmetrical, let us place the lens so that
its central plane CD is also perpendicular to LF, The line LF,
which passes through the center of the lens and is perpendicular
to the plane CD, is called the optical axis of the lens.

Since the light waves in this case constitute a parallel beam,
and since they are moving in the direction of the axis of the lens,
it is evident from the symmetry of the figure that they will be
brought together at a point on that axis such as F. Then F
will be the focus. We note that the direction of motion of the
central part of the wave has not been changed while the outside
portions have been bent through lACF. The point F, at

384

OPTICAL INSTRUMENTS 385

which parallel rays are brought together, is called the principal
FOCUS. In the case under consideration, we may measure the
distance ZF, and this is the principal focal length.

361. Image of a Point Source. If, with our lens, we form an

image of some object that is not very distant, say of a luminous

point S on the

axis. Fig. 209, the

image / of that ,

point will lie

farther away from

the lens L than ^ , ^ «

. , Fig. 209. Imagk of a Point Source

the principal focus

F, because the incident waves from a near point are not plane but

convex, and, since the thickness of the lens is the same as before,

and the light passes through its center in the same direction as

before, the same retardation will be produced in the center of the

beam. But part of that retardation is now necessary to render

the incident waves plane, and so less is left to make the plane

waves concave; therefore, they come to a focus at sonae point I

farther away from the lens than the principal focus F,

If we bring the point S still nearer to the lens, we find that its

image is still farther away (Fig. 210), and when the distance of

the point S from

• the center of the

I lens is equal to

the principal focal

length, the waves

behind the lens

Fig. 210. As the Source Approaches the Image become plane and

xvECEDES

we have a parallel
beam, Fig. 211. We see that this figure is the converse of Fig. 208.
This reciprocal relation is general in optics. // the source is placed
where the image was, the image will he found where the source was.

362. Characteristic Rays. In discussing the formation of
images of extended objects by lenses, it is simpler to consider

386

PHYSICS

the rays only, and not the waves; so we shall use rays in the re-
mainder of the explanation.

In the general discussion just given, we note that in the forma-
tion of an image two rays are particularly well defined, viz., the
one that passes through the center of the
lens, and the one that is parallel to the
axis. Since the ray that passes through
the center of the lens is not bent, its path
is determined by the point source and the
center of the lens. Since rays parallel to
the axis, after passing the lens, go through
the principal focus, this ray is determined"
by the distance of the ray from the axis,
the direction of the axis, and the principal focus. These two
rays enable us to find out many important things concerning the
relations between objects and images formed by lenses.

Fio. 211. Source at the
Principal Focus

363. Construction of the Image. For example, suppose
that we have an object 00', Fig. 212, at a distance ML in front of
the lens whose principal focus is at F, Where ^ill the image
of the object be? Its position may be found as follows: From
O draw a ray through the center of the lens L. This ray passes
through the lens without being deflected, and the image of O must
lie on this line at some point, as /. Similarly, a ray from 0' through
the center of the lens passes through the lens without being de-
flected, and the image of 0' must lie on this line at some point, as
r. Thus it appears
that the image must
lie within the angle
ILP. We shall call
this angle the lens
ANGLE of the image.
The angle OZO' will
be called the lens angle of the object, and we see that the two are
equal. The lens angle will be found to be very useful in discuss-
ing optical instruments. It is defined as the angle subtended at
the center of the lens by either the object or the image. It will

Fig. 212. Construction for the Image

OPTICAL INSTRUMENTS 387

be noted that the lens angle depends only on the size of the object
and its distance from the lens, and is independent of the size or
shape of the lens used.

In order to locate the points I and'/' on the sides of the lens
angle, from draw a ray parallel to the axis MF. This ray
must pass through the principal focus, and the image of must
lie on it. Hence the image of must lie at the intersection / of
the two rays FI and LI, When we have located the image /' of
the point 0' in the same way, it will be noted that the image is in-
verted, as was found to be the case, in the last chapter (Art. 350).
This simple construction is very useful and always gives us a close
approximation to the position of the image; for both the center of
the lens and the principal focal length can always be determined
with a fair degree of accuracy, the former from the geometrical
symmetry of the lens, and the latter by fonning an image of the
sun, as just described.

364. Size and Distance of the Image. What are the relative
sizes of the objects and the image? To understand the answer

to this question, we
must first distinguish
between angular and
.LINEAR size. If we
^i mean the former, the

Fia. 213. Relative Sizes op Object and Image ^^^^ ^* ^^^ image IS

the same as the size of
the object, i.e., the angle subtended hj the object at the center of a
lens is tlie same as that subtended by the image at the same pmnt
This fact must be carefully noted, because it is fundamental in
understanding the operation of optical instruments.

The relative linear sizes of the object and its image are differ-
ent, however, for they are at different distances from the lens.
In Fig. 213 the object OM and the image IN are both perpen-
dicular to the axis MN of the lens and their lens angles are equal,
i.e., /.OLM = /.ILN, Therefore the right triangles ILN and

OLM are similar, and j%j ^ y?J' ^^' ^^ linear size of the object

388 PHYSICS

w to the linear size of the image as the distance of the object from
the lens is to the distance of the image. These two distances arc
therefore of great importance in the discussion of relative linear
sizes. They are called the conjugate focal lengths. Simi-
larly, the points M and N, so related that an object at either of
them forms an image at the other, are called conjugate foci.

We can now determine the relative linear sizes of object and
image. We see that when M L is greater than L N the object is
larger than the image; i.e., when the object is farther from the lens
than the image, the object is larger. Conversely, when the iwxige
is farther from the lens than the object, the image is larger, and
when object and image are at equal distances from the lens they
have the same size. In this latter case the distance of the object

or the image from the lens is twice
the principal focal length of the
lens. In this position the object
and the image are as hear to-
gether as they can be. Thus we
see that as an object is brought

Fig. 214. The Object is at the au i xu •

Pkincipal Focus nearer the lens, the image re-

cedes from the lens and becomes
bigger as every photographer knows. This increase in the size
of the image is due both to an increase in the lens angle as the
object approaches, and to the increase in the distance of the
image from the lens. Is there any limit to this process? Can
we bring the object close to the lens and get an infinitely large
image? Let us see.

We have already remarked that when the distance of the
object from the lenii is equal to the principal focal length of the
lens the emergent rays are parallel. Do such rays meet? Where,
then, is the image formed? We may conceive that an infinitely
large and infinitely distant image is formed when the distance of
the object from the lens is equal to the principal focal length.
This case is illustrated in Fig. 214.

365. A Virtual Image. Suppose we bring the object still
nearer to the lens, where will the image lie? We may find out by

OPTICAL INSTRUMENTS

constructing a diagram, Fig. 215. Evidently the rays are divergent

after leaving the lens. Where, then, is their point of intersection

at which the image is formed? They

do not intersect after leaving the

lens, so that they are miable to form

an image on a screen. They do,

however, proceed as if they came

from a point behind the lens. Yet

if we place a screen aj; that point, we

have no image formed on it, for the

rays do not actually intersect there.

Hence, such an image is said to be

VIRTUAL. It will be noted that this \

virtual image is not inverted, as real images are, and that in this

case, also, the lens angle of the image and that of the object are

the same.

Fia. 215. TiiE Image is Virtual

366. How the Eye is Focused. In Art. 361 we have learned
that as the object is brought nearer the lens, the image is formed
at a greater distance from it. In the eye, however, the distance
between the lens and the retina, where the image is formed, is
constant; yet we can see both distant and near objects clearly.
Unless there were some means of focusing the image on the retina,
it would not always appear sharply defined. The device employed
is that of changing the thickness of the lens. For if (Fig. 209) the
lens were made still thicker in the middle, the central portions of
the beam would be retarded still more with respect to those at
the rim, and the curvature of the waves would be changed more
in passing through. This increased curvature would bring the
waves to a focus nearer the lens.

In order to bring the images of objects that are near-by to a
focus on the retina, the rim of the crystalline lens of the eye is
Surrounded with a small muscle which contracts, and squeezes
the lens so that it bulges out and becomes thicker in the middle.
Thus its focal length is shortened and the near-by objects brought
to a focus on the retina instead of at a point behind it. The eye
thus ACCOMMODATES itself to the different focal distances. This

390

PHYSICS

fi^^:

accommodation of the eye is limited. In a normal eye the limit
is reached when the object is about 25 cm (= 10 inches) from the

eye. If we bring the
object nearer than this,
the image recedes be-
hind the retina, and
since there is no dis-
tinct image on the
retina we can not see the object clearly. This distance — ^the least
at which the object can he, placed from the eye and still form a
distinct image on the retina — is, therefore, called the limit of

DISTINCT VISION.

Fio. 216. Far-Siohted Eye

367. Spectacles. If the crystalline lens in the eye is not normal,
it does not form clear images of objects at all distances down to
25 cm. If it is too weak, i.e., too thin in the middle, the accom-
modating muscle must be used in forming the images even of
distant objects, and will have to squeeze the lens harder in order
to focus near-by objects on the retina. Therefore this muscle
rarely gets any rest while its possessor is awake. This defect of
the eye is called far-sightedness (Fig. 216), and is likely to cause
serious results unless corrected. Far-sightedness is corrected by
strengthening the crystalline lens and making it thicker in the mid-
dle. This necessitates introducing in front of the eye a lens which
is thicker in the middle than at the' rim. This lens is indicated
by ab in Fig. 216. The strength of the added lens must be
such that distant objects are seen clearly when the accommodat-
ing muscle is relaxed.

Conversely, an eye is near-sighted when the crystalline lens
is too thick in the mid-
dle. Then the images of
objects are distinct only
when held very close to
the eye, but those of
distant objects are formed
in front of the retina (Fig. 217). This defect is corrected by plac-
ing in front of the eye a lens that is thinner in the middle than at

Fig. 217. Near-Sighted Eye

OPTICAL INSTRUMENTS

391

Fig. 218. Limit of Distinct Vision

the rim, as indicated by cd in Fig. 217. In this case, also,
the added lens should be of such curvature that distant objects
are seen clearly when the accommodating muscle is relaxed.

368. The Simple Microscope. We are now prepared to under-
stand the operation of the simple microscope. This consists of a
single lens which is thicker in the middle than at the rim. It
therefore does not differ in its action from the far-sighted spec-
tacle lens. It merely
enables us to focus ^*
clearly on an object
which is much nearer
the eye than the limit
of distinct vision. Fig.

218 shows the object at the limit of distinct
vision, and Fig. 219 shows the same object
brought nearer the eye and focused on the
retina with the help of the microscope lens.
It is clear that in this case the lens angle
OLO' is greater than before. Since our ap-
preciation of the size of an object depends on the angle that
it subtends at the eye, and since that angle depends only on
the size of the object and its distance from the eye, it is evident
that the microscope enlarges the apparent size of objects, because
it enables us to see an object clearly when it is very near the eye,
so that its lens angle there is large. This lens angle of the eye
is generally called the visual angle.

369. The Camera. Besides the eye and the simple microscope,
probably the camera is the best known optical instrument. We
have just discussed (Art. 350) the formation of an image by a small
hole in a shutter, and shown how that image is made clearer by
the introduction of a convex lens. Every one must recognize the
image formation thus produced as identical in every respect with
that of the camera. But a camera lens consists not of a single
reading glass, but of several lenses mounted together in a tube.
Nevertheless, in this case also the lens angles of the object and ot

Fig. 219. Simple
Microscope

392 PHYSICS

the image are the same, since the combination of lenses may be
replaced by a single lens that would produce a similar effect. The
discussion of all the reasons for thus making the camera lens of
several parts, would lead us far beyond the limits of our present
study. The use of the stops, in such lenses, however, demands
attention.

370. Stops. In the first place, every photographer knows that
photographic lenses are supplied with stops which limit the amount
of the lens used. The effect of a stop is twofold, viz.: 1, it reduces
the amount of light admitted to the camera and so lengthens the
necessary time of exposing the sensitive plate to the light that
comes from the object through the lens; and 2, it makes the image
on the plate sharper at the edges. The relation between the area
of the opening in the stop and the time of exposure is simple; for
the amount of light that enters the lens from a point on the object
is proportional to that area, and therefore the intensity of the light
at a point on the plate must vary as that area, other things remain-
ing the same. Hence, with a given lens, a stop whose opening has
half the diameter of the lens requires an exposure four times as
long as that required for the lens without the stop, since the areas of
the openings are proportional to the squares of the diameters.
It will be noted that the introduction of the stop does not change
the lens angle.

371. Spherical Aberration. The other effect of the stop now
demands our attention, viz., that the clearness of the image around
the edges of the plate is improved by reducing the size of the
opening in the stop. The reason for this is rather complex and
requires for its complete explanation a more exhaustive study
study than can be undertaken here. Suffice it to say, the theory
shows that lenses whose surfaces are portions of a sphere can not
bring all the points of a plane image to a focus in a plane. Thus,
if we have a plane object perpendicular to the axis at M, Fig.
220, the points of the object near the axis LN will be focused in
a plane perpendicular to the axis at N. This plane is called
the focal plane. But points of the object that are farther away

OPTICAL INSTRUMENTS

393

FiQ. 220.

The Image Does not Lie in a
Plane

from the axis, will be focused either in front of or behind this
focal plane.

In order to bring all the points of the object to a focus in one
plane, the surfaces of the lenses -would have to be portions of
ellipsoids instead of

spheres. But as ellip- l! ' " — — -..^.^^^^^ iL_-^-^ — ^^'
soidal surfaces are al-
most impossible to grind
and polish, their use is
practically out of the
question. However, the difference between a spherical and
an elliptical surface is small if we consider only a small area
of each. Therefore, when we reduce the area of the hole in the
stop, we allow only the central portion of the lens to be used, and,
therefore, we make its difference from the theoretically correct
shape very small. So the image becomes clearer. This
blurring of the image because of the spherical shape of the lens
surfaces is called spherical aberration. It has been found that
the spherical aberration of a lens may be somewhat reduced by
using several lenses instead of one, so that this is one reason why
photographic lenses are made of several parts. Another reason
will be discussed in the next chapter while studying color.

372. The Astronomical Telescope. The next instrument to
which we shall direct our attention is the astronomical telescope.
This consists of at least two lenses of the type of the simple micro-

FiG. 221. The Astronomical Telescope

scope. Inasmuch as the telescope is usually used for observing
distant objects, we shall assume that the beam of light from every
point of the object is a parallel beam. Thus, in Fig. 221, let the
line L I represent the path of the incident beam coming from the

394 PHYSICS

tip of a distant arrow and passing through the lens i, which is called
the OBJECTIVE. If F is the principal focal length of L, the image
of this point will then be at 7 in the principal focal plane. Simi-
larly, the image of a point at the other end of the arrow will be
at /'. The lens angle of the object is then equal to that of the image,
i.e., it is the angle ILI\ If we introduce a second lens behind
the image in such a way that its principal focus is also at F,> this
second lens will render parallel the light from each point of IT,
so that the beam from each point on the object is a parallel beam
when it leaves this second lens as well as when is strikes the ob-
jective.

The image IP is now the object for the combination consisting
of the second lens and the eye, and its lens angle with respect to
this combination is IL'I'. Since the image IV is nearer tp the
eye combination than to the objective, its lens angle at that com-
bination is larger than its lens angle at the objective, i.e.,
ILT > ILI'; and, therefore, the object appears enlarged. It
will be noted that the image is inverted. Since the second
lens is near the eye, it is called the eyepiece. The combination
of an eyepiece or a simple microscope with the eye will be called
the eye combination.

The reason why the telescope makes things appear larger is
now apparent. The visual angle of the object, when viewed
without the telescope, is small^ because the object is usually at a
great distance. This angle is very nearly the same as the lens
angle of the object at the objective of the telescope. But the
objective forms an image close to the eye combination. Since
this image is close to the eye combination and at a much greater
distance from the objective, its lens angle in that combination is
larger than the visual angle of the object. The magnification may
lens angle of the eye combination _ IL'T _^ lUF
lens angle of the objective IhV ILF

In Art. 6 we learned that an angle may be measured by its tangent,

IF . . IF

and that the tangent of IL'F = YTp- Similarly, tangent ILF = y-p-

rpi . ,u -fi *• • tangent lUF LF

Therefore the magnihcation is ^ — - — ; — ^^rir = -rm-
^ tangent ILF LF

Plate Vlll. The 40-inch Telescope, Yerkes Observatory,

University of Chicago ^ ^ LLl A Mb 13 A \ , VY IS.

OPTICAL INSTRUMENTS 395

But LF is the principal focal length of the objective, and L'F
is that of the eyepiece; so we see that the magnification of the
telescope is determined by the ratio of the focal length of the
objective to that of the eyepiece. Therefore, if we wish to make
telescopes that shall have large magnifying powers, we must so
construct the objective that it will have a great principal focal
length, while the eyejHece must be made to have a small focal
length.

Plate VIII is a photograph of the Yerkes telescope, which is
the largest in the worid. It has a focal length of about 25 m;
therefore, with an eyepiece which had a focal length of 0.5 cm,
its magnification would be 5000. Such a high magnification can
seldom be used, on accoimt of the unsteadiness of the atmosphere.
Since the more the image is enlarged the fainter becomes the light
in each cm* of it, it follows that when large magnificjitiohs are
used large lenses are necessary, in order to gather as much light as
possible into the image. The objective of this telescope is 100
cm in diameter.

373. The Concave Lens. If we examine a common opera glass,
we find that the eyepiece is thinner in the middle than at the rim.
Such a lens is called concave; those
that are thicker in the middle being
CONVEX. We have noted (Art. 367) the
action of the concave lens in aiding
near-sighted persons to see more clearly.
To gain a better understanding of the

.. « 1 'J i7« FiQ. 222. The Concave Lens

action ot concave lenses, consider i^ig. Scattehs the Light

222. Light waves from [the point S

spread out and fall on the concave lens L, Since the lens is thicker
at the rim than in the center, the waves that pass through the rim
are more retarded than those passing through the center. The
result is *that the divergent beams from S- are rendered more
divergent, so that the waves behind the lens will appear to come
from some point as I. Thus, / is the image of S, and it is virtual.
When the point Sis faraway, so that the incident beam is parallel,
the principal focus will be near / and will also be virtual. Con-

396

PHYSICS

versely, if we have a convergent beam that would otherwise come
to a focus at the principal focus F of a concave lens, the interposi-
tion of this lens will make the beam parallel (Fig. 223). If we place
an arrow at S and construct the image as described in Art. 363,
'"^^^^.^ we find that the lens angle of the ob-

;; ^ ject is equal to that of the image, i.e..

OLO' = ILP, Fig. 213.

Fig. 223. Parallel Beam
Formed by CJoncave Lens

374. The Opera Glass. Let us now
construct a diagram to ^present the
visual angles as they occur in the opera
glass. The objective alone would form, an inverted image //' of
the distant object at its principal focus F. The lens angle of the
object is, then, equal to FLL The concave lens is introduced in
front of the image, with its principal focus also at F. Then, since
the rays which fall on i' from any point of the object are converging
toward a point in the plane of its principal focus, they will be ren-
dered parallel by the eyepiece (cf. Art. 373). The direction of each
such parallel beam will be that of the corresponding point of the
image //' from the center of the lens U, Hence, the lens angle

Fig. 224. The Opera Glass

of the image after passing the lens U will be IL'I; and there-
fore, as in the case of the astronomical telescope, the magnification
is the ratio of the focal lengths of the objective and the eyepiece.
For viewing ordinary objects this instrument has an advantage

OPTICAL INSTRUMENTS

397

over the telescope in that the image formed by it is upright.
The opera glass is often called the Galilean telescope, since it
is the kind invented by Galileo and with which he discovered the
satellites of Jupiter.

375. The Compound Microscope. This instrument differs from
the telescope only in the fact that the object is placed in front of
the objective near its principal focus, so that its lens angle
may be made as large as possible. A real image is formed by
the objective near the eye combination, so that the lens angle of
this real image with respect to that combination is also large.
Fig. 225 shows the arrangement, ILI' being the lens angle of the

Fig. 225. The Compound Microscope

object, and IW the lens angle of the image with respect to the
eye combination. By using an objective of very short focal
length, real iinages may be obtained when the object is brought
very close to the objective. But, provided it remains outside of
the principal focus, the nearer the object is brought to the ob-
jective, the larger is the lens angle of the object, and also the
larger i? the linear size of the image. Consequently, by this ar-
rangement very large magnifications may be obtained, the only
limit being the intensity of the light; for as the image gets larger
it becomes fainter. When we reach a magnification of about
2500 we have reached the practical limit for eye work. Photo-
graphs can, of course, be made with still larger magnification3,
but the exposures must be very long.

One interesting point may yet be mentioned. Although we
can magnify an object without limit, a point is eventually reached

398 PHYSICS

beyond which greater magnification fails to reveal further detail
in the object.- What is that limit? Though we can not here give*
the reasons for the conclusion, we may, nevertheless, state it.
When the distance between two points of an object is less than
1-100,000 of an inch, we are not able, by any known optical device,
to distinguish whether there are two points or only one. Thus,
when an object has been magnified imtil points one-hundred-
thousandth of an inch apart are separated, further magnification
will not reveal any further details of the construction of the object.
This point is of interest, because the ultimate particles of matter —
atoms and molecules — ^are much closer together than this in solids,
and therefore we know that we can never see them with our
eyes, though we may be able to know them in other ways.

376. The Photometer. In the discussion of the astronomical
telescope and of the microscope, we have found that the intensity
of illumination of the image is a matter of importance. In prac-
tical life it is a matter of even greater importance, since all artificial
lighting by gas, or electricity, is measured and rated according to
its intensity. The unit in which intensities are measured is the
CANDLE-POWER. This is the rate at which light is radiated from
a candle of specified construction burning a specified amount of
sperm per minute. The ordinary electric glow lamps are generally
equivalent to 16 standard candles, and are therefore called 16
C.P. (candle-power) lamps.

Intensities are compared by means of a photometer. The two
lights to be compared, e.g., a lamp and a standard candle, are set
about 2 or 3 m apart, and a piece of paper with a grease spot "on
it is supported between them. This paper is moved backward or
forward until the spot can no longer be seen. When this is the
case, the illumination of the two sides of the paper is the same.
The intensities of the two lights must be directly proportional to
the squares of their distances from the paper. If the distance
of the standard candle from the paper is 20 cm and of the lamp
80 cm, or 4 times that of the candle, then the intensity of the lamp
is 4* times that of the candle, or 16 C. P. This may be proved
by considering that the light is spreading out in all directions

OPTICAL. INSTRUMENTS 399

from each source, so that the energy that spreads over 1 cm'
on a surface 1 m from the light is spread over 4 cm^ on a surface
twice as far from the light. This form of photometer is called the
Bunsen photometer, after its inventor, and as we have just seen,
it is based on the principle that the amount of light which falls
from a given source on ea>ch cm* of a surface is inversely proportional
to the square of the distance of the source from the surface. This is
a geometrical consequence of the straight paths of the rays; but
it is strictly true only when the distance is large compared with
the size of the source, so that the light may be regarded as diverg-
ing from a point.

SUMMARY

1. When parallel rays are brought to a focus by a convex lens,
the distance from the lens to the focus is called the principal focal
length.

2. As the object is brought nearer the lens, the image recedes
from it. When the image is real, it is inverted.

3. When an image is formed by a convex lens, the ratio of
the linear size of the object to that of the image is equal to th:)
ratio of the distance of the object, from the lens, to that of the
image, from the lens.

4. When the distance of the object from a convex lens is less
than the principal focal length, the image is erect and virtual.

5. When an object and its real image have the same size, the
distance of each from the lens is equal to twice the principal focal
length.

6. No real image can be formed by a convex lens when the
object and the screen are nearer together than four times the
principal focal length of the lens.

7. The eye is focused by changing the thickness of the crys-
talline lens.

8. A normal eye can not form on the retina clear images of
objects that are nearer to the eye than 25 cm. This distance is,
therefore, called the limit of distinct vision.

9. An eye is far-sighted when the accommodating muscle
must be used to see distant objects clearly. It is then unable to

400 PHYSICS

bring the images of near objects to a focus on the retina without
straining the acconunodating muscle.

10. An eye is near-sighted when its crystalline lens is too thick
in the middle. It cannot bring the images of distant objects to a
focus on the retina.

11. When an image is formed by a lens, the lens angle of the
object is the same as that of the image.

12. The simple microscope enables the eye to focus cleariy
on the retina the images of objects that are less distant from the
the limit of distinct vision. The object appears enlarged, because
it then subtends a larger visual angle.

13. In the telescope and the opera glass, the light from each
point of the object is parallel when it enters the objective, and
also parallel when it leaves the eyepiece. The magnification is
due to the fact that the real image formed by the objective is
nearer to the eye combination than it is to the objective, so that
the lens angle of this image with respect to that combination is
larger. The telescope and the opera glass therefore give the
same effect, enlarging the visual angle of the object.

14. A concave lens causes the light rays to diverge, and forms
only virtual images.

15. The practical unit of light is that furnished by a standard
candle.

16. The intensity of the light that falls on 1 cm^ of a surface
is inversely proportional to the square of the distance of the sur-
face from the source of light.

PROBLEMS

1. If you wish to copy a photograph full size with a lens of 15 cm
principal focal length, how great must be the distance between the
picture and the photographic plate? What will be the relation between
the distances from the lens to the object and image respectively? If
you wish to enlarge the picture to twice its. size, what will be the rela-
tion between those distances? If you wish to get the number of cm
in this case, you must know the relation between the two conjugate
focal lengths and the principal focal length. If u represent the dis-
tance of the object from the lens, v that of the image, and / the

OPTICAL INSTRUMENTS 401

principal focal length, the relation among the three is found to be

i._j_l. = L. Verify this equation in the case of object and image the

u V f

same size (w = v = 2/). ;

2. Draw a diagram of a convex lens with a principal focal length
of 4 cm. Draw an object 6 cm from the lens and construct the image
by the method of Art. 363. What is the relative size of the image?
Measure the distance from the lens to the image and see if the construc-
tion verifies approximately the equation in problem 1.

3. A 16 c. p. incandescent lamp and an arc lamp give the same
illumination on a screen when the distance from the screen of the in-
candescent lamp is 10 cm and that of the arc lamp is 100 cm; what is
the candle-power of the arc lamp?

4. If the arc lamp in problem 3 takes 9 amperes at 55 volts,
while the incandescent one takes 0.5 amperes at 110 volts, at what
rate in watts is energy supplied to each? Which lamp is the more
efficient? What is the ratio of their efficiencies?

5. The amount of light that -passes through a camera leps is pro-
portional to the area of the opening of the lens, i.e., to r^, if r is the radius
of the opening. If we have two lenses with diflferent sized openings
of radii, r^ and r2, and if i^ and t2 represent the intensities of illumination
of the light on a cm2 of the plate at a fixed focal distance /, write the
proportion which expresses the relation between the intensities and the
radii of the openings

^. If we have two camera lenses of the same area of opening but
of .different focal lengths /i and /2, the same amount of light will pass
through each under like conditions, but since the focal length of one
is greater than that of the other, the intensity of illumination of the
light on one cm^ of the plate will be inversely as. the squares of the
focal lengths, i.e., ii : ij = /| : f\. Does this explain why a long focus
lens is "slower" for taking pictures than a shorter focus one of the
same aperture?

7. Multiply together the two equations of problems 5 and 6 and

extract the square root. The result is i : 12 = ^ : ^. What angle is

measured by ^? Twice this angle, or that subtended by the rim of the

/I
lens at a point on the plate, is called the angle of aperture. Since
the "speed" of a photographic lens depends on the intensity of light
on the plate, may we use this ratio, or its double, the ratio of the diam-
eter of the lens to the focal length, as a measure of speed? What is

the meaning of the marks ^, etc., on the stops in some camera lenses?

402 PHY8ICJ8

SUGGESTIONS TO STUDENTS
1. If you have a camera, measure the diameter of the stop marked

and then measure the focal length of the lens — preferably by forming

an image the same size as the object — ^and dividing the distance be-
tween object and image by 4. Do you find any relation between the
things measured and the numbering on the stop?

2. Take your opera glass lenses out, and measure the focal lengths
of the objective and eyepiece. This may be done by letting sunlight
pass through each and measuring the distance at which the sun's image
is formed by the objective. The eyepiece spreads the light, but its
focal length may be found by measuring the diameter of the lens, and
the diameter of the spot of light formed by it on a screen at a measured
distance. What is the magnification? Determine it by looking with
one eye at a brick waU and with the other at the same brick wall through
one tube of the opera glass; you then see two images of the bricks, one
without the glass the other with it; the magnification is the number of
bricks in the first image which cover one in the second. You can
also determine the magnification of telescopes in this same way. Would,
a plainly marked scale of equal parts be better than the brick wall?

3. Examine the projecting lantern in the schoolroom and see if
you understand the operation of its "condenser." Is its projecting
lens different from a camera lens? Measure the distances from slide
to lens and lens to screen, and see if the diameter of the slide is to that
of the image on the screen as the respective distances are to each other.

4. There are many useful books on photography. You will also
find Wright, Optical Projection (Longmans, New York), valuable.
There are some interesting experiments in optics in Hopkins's Experi-
mental Science. , The best book on light for young students is Professor
S. P. Thompson's Light, Visible and Invisible (Macmillan, New York).
This is a series of the Christmas Lectures at the Royal Institution in
London. It gives a very clear and fascinating account of the latest
experiments and theories. See also Tyndall's Six Lectures on Light,
which gives an account of the work of Newton and Young. Tyndall's
style is a model in clearness and precision, and the charm of the man
is reflected from every page.

CHAPTER XXI

COLOR

377. Newton's Experiment. Having studied in the last two
chapters the way in which light serves us by enabling us to dis-
tinguish differences in direction, we shall now pass on to the
discussion of color phenomena. Although color has been ob-
served, and used extensively from time immemorial, little was
known concerning the reasons why different substances appear to
have different colors, until the time of Newton. In 1675, Newton
discovered that a beam of sunlight when admitted through a small
hole H insi shutter into a dark room and then sent through an ordi-
nary glass prism P, was not only bent from its path, but was
also spread out into a band of various colors, extending from red to
violet (Fig. 226). He further found that if a second prism, in all

Fio. 226. Newton's Experiment

respects like the first, was introduced behind the first in such a
way that the two together made a thick plate of glass with parallel
sides, the colored band was reduced to a colorless spot. From
these experiments Newton concluded that white sunlight is com-
posed of all the colors of the rainbow, and this conclusion has
been verified by further investigation.

Now, this experiment shows that white light is a mixture of
all different colors in certain proportions, but it does not tell us

403

404 PHYSICS

wherein the colors differ from one another. What is the physical
difference between lights of different colors; for example, between
red light and blue light? To this question Newton gave no sat-
isfactory answer, because he did not conceive light to be a wave
motion. But when we adopt the theory that light consists
of waves, we are able to form a clear conception of the physical
nature of differences in color, viz., that the different colored lights
correspond to waves of different lengths, just as sounds of different
pitch correspond to waves of different lengths. How can we
prove that this is so; i.e., how can we measure the lengths of the
waves of light, in order to find out if they are different for different
colors?

378. Interference Fringes. This may be done in a number of

ways, but probably the simplest is the following: Carefully clean

two pieces of the best plate glass, and clamp them together

so that they touch along one edge and are held apart by

a fine hair or fiber at the other (Fig. 227). We thus

have formed between the two plates a wedge of air. If

now we allow light of one color, like that from a flame

colored with salt, to fall perpendicularly on these plates,

we do not see the familiar image of the flame, but,

instead, there appears a series of bright bands of equal

width, separated by dark spaces, which are also of equal

width. These bands are called interference fringes, and

Am they plainly show that there is something periodic about

the light, just as the phenomenon of beats makes evident

the periodic nature of sound.

No very satisfactory reason can be given for the appearance
of the black bands unless we conceive that the light consists of
waves. We have learned in the chapters on wave motion and
sound that two waves may add themselves together so as to pro-
duce no motion when their phases are opposite. Similarly, in the
case of the two plates of glass, we may suppose that waves of
equal length, but in opposite phases, are thus adding themselyes
together so as to destroy each other's effects, therefore darkness

COLOR

405

results. Whence do we get the two waves? By reflection from
the inner surfaces of the two plates. Thus, let PQ and PR,
Fig. 228, represent these two surfaces^___WSen-the^
light falls on the surface PQ, part of iFls^T^flected
and part passes through. When that which has
passed through falls on the second surface •Pii,
part of it is reflected and thus we have two re-
flected beams, one a d from the first surface, and
the other b e from the second. Now, it is clear
that the light reflected at b will, when it reaches c,
be somewhat different in phase from that reflected
at a, because the light at c will be some wave
lengths behind that reflected at a, since it has
traveled a distance abc more than the light at a.
Therefore, if this extra distance is half a wave, the
light at c will be half a wave behind that at a, and
so the two beams will come together in opposite phases, cancel
each other's effect, and together produce darkness (c/. Art. 299).
Similarly, when the extra distance abc is a whole wave, the two
rays ad and ce will be in the same phase, and so when they are
added together, they produce light. Therefore, at that place we
see a bright band crossing the glass. When
the distance abc is three half waves, dark-
ness again results, and so on. Therefore the
successive bright bands occur at places
where the successive extra distances abc
traveled by the second beam differ by a
whole wave, and that the dark bands occur
when these distances differ by half a wave.
Thus, in Fig. 229 the bright bands are
found at the points marked 6, and the dark
ones at the points marked d. It is clear
from the figure, that between two bright
bands the distance from plate A B to
plate C D increases by one-half of a wave.
This distance can be measured by deter-
mining the diameter of the fiber, its distance from the edges of

Fio. 229. Interference

OP Waves in the Air

Wedge

406 PHYSICS

the plate, and. the distance between the dark and bright bands.
For example, if the diameter of the fiber is 0.01 mm, and if it is
60 mm from the place where the plates touch, and if the distance
between the bands in red light is 2 mm, we may see 30 bands
on the plate. Since in this case 30 bands correspond to a change
in distance between the plates of 0.01 mm, one band corresponds
to a change of ^^^^ = .00033 mm. But this change in distance
corresponds to half a wave length of red light; therefore
2 X .00033 = .00066 mm = one wave length of red light.
There are several other ways of determining the lengths of light
waves, all of which are vastly more accurate than this. They
are all based on the principle of interference, and the values
of the wave lengths obtained by the different methods agree
closely with one another.

379. Lengths of the Waves of the Colored Lights. If we now
illuminate this same pair of plates with green light, the dark bands
will appear narrower and nearer together — there will be about 40
on the plate. Therefore the distance between the plates in-
creases f ^ for each band, i.e., .00025 mm. Therefore one wave
kngth m green = .00025 X 2 = .00050 mm. Proceeding in a
similar manner for the other colors, we find the lengths of the
corresponding waves to be approximately:

red, .00066 mm

orange, .00060 mm

yellow, .00055 mm

green, .00050 mm

blue, .00045 mm

violet, .00040 mm

We thus find that different colors really do have different wave
lengths. It is interesting to note how extremely minute the waves
are. Thus, there are about 2500 blue waves or 2000 green ones,
or 1500 red ones in one millimeter or 38,000 red waves in an inch.

380. Interference Fringes in White Light. What will happen
if we allow white light, instead of light of one color, to fall on the

COLOR 407

two glass plates? We find a most beautiful array of many-colored '
bands. The colors are not so marked, however, as those in the
spectrum. These bands are easily produced with an ordinary
soap solution, such as is used for blowing bubbles. A drinking
glass is dipped in the soap suds and then set on its side, so that
the soap film over its end is vertical. As the water drains out
from between the two sides of the film, a thin wedge of soap solu-
tion is formed, and the colored bands will be seen stretching hori-
zontally across the film. The hypothesis that light consists of
waves enables us to give a simple explanation of the formation
of these colored bands. For if all the colors are present in white
light, and if the different colors correspond to waves of different
lengths, then each color will have a set of bands corresponding
to it, and these bands will be of different widths. When these
different sets of colored bands are all present, they overlap, so as
to give us color mijctures. These mixtures of the different colors
produce the various tints or tones observed in the two glass plates
or 'the soap film.

Thus one of the mysteries of our early childhood is solved;
for the colors often seen in a crystal or a piece of ice which has a
crack in it, are formed in this way. So, also, are the colors in a
soap film, or those seen on the surface of oily water. It is by
reflection from the two surfaces of the bubble, or the crack, that
these wonderfully colored interference bands are produced. The
iridescence of polished shells, and of certain kinds of glass, may
be accounted for in a somewhat similar manner by the interference
of waves reflected under suitable conditions from their surfaces.

381. Dispersion. Having thus learned that different colors cor-
respond to different wave lengths and that white light is a com-
position of all the colors, let Us return for a moment to Newton's
experiment with the prism, and ask how the prism is able to sepa-
rate the colors and spread them out into a band. Clearly, the
prism must be acting differently on the different colored
lights. If we repeat Newton's experiment, and interpose a red
glass in the beam of sunlight, we find that the path of the red light
- Js bent a certain amount by the prism, and we get on the screen a

408 PHYSICS

red spot only. On changing the red glass for a blue one, we
observe a spot on the screen, but the blue spot does not fall on the
screen at the same place as was occupied by the red one. On
examination we find that the prism changes the direction of the
blue light more than it does that of the red. Thus the prism
separates the white light into a series of spots of color and the
entire band of color is thus seen to consist of a series of spots of
color, each overlapping the adjacent ones.

The bending of the light is due to the refraction of the glass
of the prism, and the amount of that refraction is measured by
the index of refraction, which is the ratio of the velocity of light
in the glass and in the air (cf. Art. 354). Hence, since the index
of refraction of blue is found to be different from that of red,
we conclude that the velocity of blue light in the glass differs
from that of the red. Since the blue is bent more than the red
its index is greater, and therefore its velocity in glass must
be less than that of the red. Thus, we conclude that the separa-
tion of white light into colors by the glass prism is due to the fact
that the waves of the different colored lights travel with different
velocities in glass. This phenomenon of the separation of light
into colors by a prism" is called dispersion. It is of interest to
find out whether all transparent substances separate light into
colors to the same extent that glass does, i.e., how this separation
depends on the substance of the prism.

382. The Spectrum. • It would not lead to accurate results if
we tried to measure dispersion with the apparatus as used by New-
ton, because each color in the white light forms a spotof color at a
definite place, and these spots overlap to form the band of colored
light. Since each spot is larger than the hole (cf. Art. 350), the
band is fuzzy and not clearly defined. Hence, we must so improve
the apparatus that we may measure the position of a given spot
with some degree of accuracy. As you may have surmised, this
is done by introducing a convex lens L, Fig. 230, in such a way
that it forms on the screeen images of the hole in the shutter.
Another improvement is to replace the round hole in the shutter
by a narrow slit parallel to the edge of the prism, because then the

COLOR

409

successive colored images of the hole do not overlap as much as
do the round images, as shown in the figure. When we make
these changes we find that the appearance of the band of color,

Fig. 230. The Spectrum is Clear and Brilliant

the SPECTRUM, as it is called, is much improved. The edges are
now clearly defined and the colors in the center are both brighter
and more distinct.

383. Bright Line Spectra. However, one other question needs
settlement before we can compare dispersions, viz.: What colors
shall we compare; for the spectrum appears to contaii^ all sorts
of reds, greens, etc.? We must, then, select some particular ones
for comparison. W^hich shall they be? If, with the new arrange-
ment of the apparatus, we interpose a red glass, we find that even
now the spectrum of the transmitted light is not sufficiently well
defined to admit of definite measurement; for the red spot, though
rectangular, is broad and contains many different shades of red,
i.e., is not monochromatic. Nature has furnished us with a means
of producing suitable monochromatic colors, for we find that if
we burn a metal, sodium, for example, the light is nearly mono-
chromatic. Thus, if we send the sodium light through our appa-
ratus — a SPECTROSCOPE, let us call it — we note that the spectrum
consists mainly of a clearly defined bright yellow image of the
slit through which the light passed before* entering the prism.
Since this image is a clearly defined yellow line, w^e may measure
its position with great accuracy. In a similar way, if we burn
mercury, we find that the corresponding spectrum consists of a

410 PHYSICS

bright yellow line, a bright green line, and a violet line. Zipc,
in the same way, gives a spectrum consisting of a red, a green,
and two blue lines. Other metals give still other lines.

Now, the great advantage of these lines is that they are very
clearly defined, and, so far as we have been able to detect, they
always have the same colors; i.e., the wave length corresponding
to a given line is always the same in air. Therefore they furnish
convenient points of reference by which to measure dispersion.

384. Measurement of Dispersion. We are now in a position to
compare the powers of two different prisms. Let us take, for
example, one of water and one of glass, both of the same size.
We first place the water prism in the spectroscope, and using
mercury light, for example, mark on the screen the positions of
the yellow, the green, and the violet lines. We then do the same
with the glass prism, and find that it not only bends each of the
rays more than the water does, but also that the distance between
the successive lines is greater. Thus, the glass not only gives the
rays a greater deviation, but also the dispersion of the rays is
greater.

Another complication now arises, for if we turn the prism
about a vertical axis, we find that both deviation and dispersion
vary; i.e., both depend on the angle of incidence of the light on the
prism. It will be found, however, that there is one position of
the prism for which the deviation is smaller than for any other.
Hence we may compare prisms when in this position, which is
called that of minimum deviation. On setting the prism in this
position, it will be found that the light enters and leaves it at the
same angle with the front and rear faces respectively. We may.
then, compare the powers of prisms when they are in this par-
ticular position.

385. Achromatic Lenses. The importance of this discussion
of deviation and dispersion becomes manifest when we apply the
principles we have learned to lenses. For it must be clear that if
the different colors travel with different velocities in glass, then
when white light passes through a lens, some of the colored beams

COLOR 411

will be refracted more than others by the lens, so that the different
colors will not all come to the same focus. Thus, if the object is
a star on the axis of a lens, the blue w^aves, being much more
retarded by the lens than the red, will come to a focus nearer the
lens than do the red waves. Hence, when we wish to observe the
image of a star, we find that there is, strictly speaking, no image,
for at one point there is a red image surrounded by fuzzy blue
light and at another a blue image surrounded by fuzzy red light,
and so on for other colors. This phenomenon is called chro-
matic ABERRATION. It thus becomes clear that unless we can find
means of correcting this aberration, lenses for fine work are useless.

Since chromatic aberration is a result of dispersion, it is clear
that the solution of the difficulty must be sought in a study of that
phenomenon. We may then ask, Do prisms of the same size, but
made of different substances, always differ in the deviations which
they produce, and also in their dispersion? Or may we have two
prisms that have the same index of refraction for some one color,
and yet have different dispersions? The answer to this question
is, of course, only obtained from careful investigation of numerous
cases. The result of such investigation is that substances have
been found which produce the same deviation, and yet have dif-
ferent dispersions.

In order to understand how this fact may be used to produce
colorless images, we must first remember that deviation alone
is what we need for this purpose. Therefore we must find two
prisms of such nature that they produce deviation without color.
This is done in a manner similar to Newton's method of recom-
bining white light by putting two prisms together with their angles
in opposite directions (c/. Art. 377). But in Newton's experiment
the two prisms were of the same size and of the same substance.
Therefore, by introducing the second prism, he canceled not only
the dispersion of the first, but also its deviation. But suppose
the second prism is of some other substance, and of such an angle
that it possesses the same dispersive power as the first but less
deviating power. Then the dispersions of the two ^vill cancel
each other and leave a colorless beam, while some of the deviation
produced by the first prism will remain.

412 PHYSICS

This method is the one actually used in constructing lenses.

If you take out a telescope or opera glass objective, you will find

that it consists of two lenses, one convex, made of crown glass,

and the other concave, made of flint glass, as shown in

Fig. 231. These two lenses are. so shaped that their

powers of dispersion are equal and that they act in

opposite directions, while the deviating power of the

convex one is greater. Therefore, in the combination

the chromatic aberration is corrected, but the lens is

still able to deviate the rays, £^nd so it forms an image

that is clear and free from the fuzzy halo of color.

Fig!^i. We owe the solution of this problem to John Dollond,

'matIc an English optician, who produced the first achromatic

^^^* objectives in the year 1757. Newton had been unable

to find suitable glasses for the constructon of such a lens, and

had therefore thought that it was impossible. By the discovery

of new sorts of glass modem investigation has shown that it is

possible to produce other types of achromatic lenses than the

one just described.

386. Spectrum Analysis. Having thus seen how the phenome-
non of dispersion nearly prevented us from constructing lenses that
were useful for fine work, let us ask if dispersion is truly useful
in any way. To this we must answer, yes, indeed; and bring
forward the following explanation in justification of the answer.
We have noted, in what has just preceded, that the light from
burning sodium, mercury, or zinc, when analyzed in the spectro-
scope, is found to consist of certain well defined colors. Each of
these colors is pretty nearly pure, i.e., it consists of waves not
differing much from one another in length. Further, the colors
emitted by each substance are found to be different, so that each
element, when burned, produces in the spectroscope a series of
bright lines that are characteristic of it. Therefore the spectrum
furnishes us with a means of determining the nature of a sub-
stance. For if we bum a substance and analyze with a spectro-
scope the light emitted, we can recognize the lines and so tell
what the substance burned is. Thus the dispersion of prisms

COLOR 413

furnishes us with a powerful and accurate method of chemical
analysis.

In this use of the spectrum we are not confined to the study
of substances of the earth only, for we receive light from the sun,
the stars, and other heavenly bodies. Because their light can be
analyzed in this way, we are able to discover the composition of
these bodies. So the dispersion of prisms enables us to extend
our chemical analysis to the farthest visible regions of the uni-
verse. The results of such study are most interesting, for we find
that many heavenly bodies produce spectra consisting of bright
lines, while others produce spectra that are of the same nature
as that of our sun. So we are able to find out that some of the
heavenly bodies are composed of burning metals, while others
are more like our sun.

387. Continuous Spectra. What is the difference between
a spectrum consisting of bright lines, and one similar to that of
the sun, which seems to contain all possible colors? We can find
out by experiment; for if we heat a metal, say a piece of zinc, in
a flame, it grows gradually hotter. As we have learned in Art.
153, it sends out long heat waves at all temperatures, and as it
grows hotter, shorter waves are added to the complex mass of
waves, until, when the temperature reaches about 520° C, it begins
to send out red light waves. If its light is then passed through
the spectroscope, the spectrum will be found to contain mainly
red waves. But as the temperature increases, the shorter waves
appear, until the entire spectrum is produced on the screen. The
zinc is then said to be white hot. But it has not yet melted. On
examining the spectrum thus formed, it will be found to be con-
tinuous, i.e., to contain all the wave lengths that correspond to
the different visible colors.

But if the zinc is heated still further until it catches fire and
bums, i.e., forms a luminous vapor, the spectrum will suddenly
change to one consisting of the bright lines characteristic of the
zinc. Thus it appears that incandescent solids produce con-
tinuous spectra, while incandescent vapors are the source of the
bright line spectra. Applying this fact to the spectra of the

414 PHYSICS

heaveniy bodies, we see that if those spectra consist of bright lines,
we are justified in concluding that the heavenly bodies that pro-
duced them consist of incandescent vapors; while if the spectrum
appears continuous, the body must be more like an incandescent
solid.

388. Dark Lines in the Spectrnm. Is our sun a vapor or a
solid? On examining the solar spectrum carefully, it will be
found to be neither continuous nor yet to resemble a bright line
spectrum; for though it appears continuous when the dispersion
used is small, a higher dispersion reveals the fact that it is crossed
by a large number of black lines. What do these black lines mean?
Evidently, that certain wave lengths are not present in the solar
light when it reaches us. A comparison of the position of these
dark lines wdth those emitted by burning metals shows that the
position of the two sets in the spectrum are the same. Thus, for
example, there are found in the solar spectrum black lines at the
same positions that are occupied by the bright lines of burning
sodium. Similarly, for the other metals. What, then, can be the
nature of the sun that it sends us all other vibrations excepting
those that the substances we know on earth emit? Is the sun made
of substances that are totally different from those of the earth?

The explanation of the presence of the dark lines in the solar
spectrum was first given by Kirchhoff in 1859, and rests on the
principle of resonance explained in Art. 314. We there learned
that bodies that emit powerfully waves of a certain period are
set into violent vibration when waves of the same period are im-
pressed on them. We have found a marked example of this in
sound (c/. Art. 342), for there we discovered that a body may be
set into violent vibration by resonance, when its natural period
of vibration agrees with the impressed period. Further, when a
body is thus set into vibration by resonance, it absorbs the radiant
energy that falls on it, begins vibrating, and thus becomes itself
a new source of waves w^hich spread out in all directions about
it. Thus, the energy of the waves which are traveling in the direc-
tion of the resounding body is scattered by that body in all direc-
tions, and therefore less of it is passed on in the original direction.

COLOR 415

Now, KirchhoflF reasoned in a similar manner concerning iight
waves. For he said, ''Since the sodium particles have natural
periods of vibration which correspond to the bright lines pro-
duced by their radiations in the spectrum, they must be able to
vibrate by resonance whenever they are acted on by waves whose
period agrees with their own. Further, when they are thus set
into vibration by resonance they must absorb the energy of the
incident waves, and scatter it in wave motion in all directions.
Therefore the black lines in the solar spectrum, which indicate
the absence of the waves that agree in period with those emitted
by sodium, show that there are, between us and the source of the
solar light, particles of sodium which absorb the sodium vibra-
tions from the sunUght, and scatter them in all directions.*'

389. The Suii'9 Atmosphere. Where can such particles be?
They do not exist in the earth's atmosphere. They do not exist
free in the cold regions of space, since they must be in the form
of a hot vapor in order to execute their free vibrations. They
must, therefore, be in the solar atmosphere. So we reach the
conclusion that sodium vapor is a constituent of the sun's atmos-
phere. Reasoning in like manner, we conclude that the other
elements, whose bright lines coincide with dark lines in the solar
spectrum, must exist as vapors in the solar atmosphere. Careful
study of the bright lines of elements and of the dark lines of the
solar spectrum shows that almost every known bright line has a
corresponding dark line there; and, therefore, we conclude that
the atmosphere of the sun is composed, not of substances different
from those in the earth, but of the same materials. The lines
in the spectra of the stars also agree in position with those of
substances known here. So we conclude that all bodies in the
visible universe are composed largely of the same substances as
are found on the earth.

390. Complex Colors. In Art. 379 we found that colors
correspond to the lengths of the light waves, and that simple
colors are those that are produced by waves of a definite length.
We have learned that incandescent solids send out white light

416 PHYSICS

consisting of all imaginable visible colors, while an incandescent
vapor sends out certain simple colors- characteristic of it. We
have seen how particles of definite natural periods of vibration
absorb and scatter the energy of waves whose periods of vibration
agree with their own. We are now ready to enter* on the investi-
gation of the reasons for the appearance of color in the ordinary
objects about us. Nothing is more familiar to us than that differ-
ent objects, illuminated by the same sunlight, appear to have
different colors. Why do the leaves appear green, the violets
blue, the goldenrod yellow, etc.?

The complete answer to this question is at present unknown.
However, in the light of the principles just learned, and with
the aid of the spectroscope, we can give a partial answer. If we
pass sunlight through the spectroscope, and then interpose various
pieces of colored glass in the path of the light, we note that the
resulting spectra are all different. Thus, when red glass is
introduced, the yellow, green, and blue of the spectrum are ab-
sorbed, while the red and some orange come through. When
green glass is introduced, the red and the blue disappear, while
the green and some yellow come through. It will, however, be
noted that this soii of absorption differs from that which produces
the dark lines of the solar spectrum. For in the case of absorp-
tion by metallic vapors, very definite colors are absorbed — what
we have termed simple colors, which correspond to one particular
wave length. But the colored glass absorbs a vast number of
wave lengths and lets another vast number pass through. So
we see that though the color of the glass, red, for instance, is due
to the absorption of part of the waves in the sunlight, yet a large
number of different waves are transmitted, so that the light
which is thus sifted by the glass is still very far from simple.

391. Colors of Ordinary Objects. We thus see that the
colors of ordinary objects are complex in the same way that sounds
are, in that both consist of a large number of different waves.
There is, however, this distinction, that the various notes in a
complex musical sound are related by simple numerical relations,
while among the waves that produce ordinary colors, no such

COLOR 417

relations exist. Consequently we have no color scale, i.e., we
have no standard universally accepted as to harmony and discord
of colors. Many attempts have been made by metaphysicians
to construct a color scale which should correspond to the musical
scale; and Newton's statement that the spectrum consists of the
seven colors, violet, indigo, blue, green, yellow, orange, red, has
frequently been used for this purpose. We are now able to see
that such attempts are entirely artificial, since there is no simple
numerical relation between the component parts of a complex
color, and so our judgment of harmony and discord of colors has
no such physical basis as has our recognition of tonal relationship
in music.

392. The Eye. We can, perhaps, make this clearer if we
compare the mechanism by which the eye detects colors with that,
by which the ear detects sounds. Experiments in the reproduc-
tion of colored pictures have shown that most of the colors recog-
nizable by the eye can be produced by combinations of red, green,
and blue light. This fact has led to the theofy of Young and
Helmholtz, that there are on the retina of the eye three sets of
nerves, one sensitive to red, one to green, and one to blue. These
three nerves must be of such a nature that exact coincidence
between their natural period and the impressed period is not
necessary in order to make them respond to the action of the waves.
Our perception of the colors, then, probably depends on the rela-
tive intensity of the excitation of these three sets of nerves. The
ear has a large number of nerve fibers, each tuned to a different
note, therefore it resolves complex tones into their components
and hears the components separately. The eye, on the other
hand, has three different sets of fibers, of which a large number
are tuned to red, another large number to green, and still another
to violet blue. Therefore red, green and violet lights, acting
together stimulate all three sets of nerves and give the sensation
of white. Blue and yellow together do the same, while green
and violet lights together give peacock blue, a tint between green
and violet, red and violet give purple, red and green give yellow,
.and so on.

418 PHYSICS

393. Paints and Dyes. The action of paints and dyes is
similar to that of colored glass in one way, but different in another.
It is different in that the light we receive from painted or dyed
surfaces is reflected, not transmitted. It is similar, in that the
light that comes to us after reflection is w^hat remains of the inci-
dent light after some of its colors have been absorbed. Thus, red
glass absorbs most of the blue, green, and yellow from white light,
and transmits only the red and some of the orange. In like man-
ner, red paint absorbs most of the other colored lights, and returns
only the red and orange by reflection.

That this is the action of paints and dyes is shown by the fact
that the colors of substances appear so different when viewed by
gas light and in the daylight. For daylight contains all the differ-
ent colors in large amounts, while gas light lacks much of the
blue and violet. Since the pigments can send back only certain
waves from among those which they receive, the blues and violets
appear dvM in gas light; and in a red light they appear nearly or
entirely black, because they absorb all the reds, and there are in
the red light no waves that they are able to reflect.

394. Mixing Lights and Mixing Pigments. Why is it that
if we mix two colored lights, say blue and yellow, we see a white
mixture on a white screen; but if we mix blue and yellow paints
or dyes, and color the screen with the mixture, the result is green?
The answer is easily obtained by considering carefully the action
of the painted and the unpainted screen on the light. The screen
is white in daylight, because it receives and reflects a mixture of
all colored lights. If we send two beams of colored light, one
yellow and the other blue, to the same point on the screen, both
beams are reflected and make their impressions at the same place
on the retina of the eye. The result is that all three sets of nerves
at that place in the eye, the red, the green, the blue, are sufli-
ciently excited to give us the impression of w^hite.

The sensation arising from the light reflected by a mixture
of blue and yellow paint is green, because the blue pigment ab-
sorbs all the incident light except the green and the blues, which
it reflects; while the yellow paint absorbs all but the orange.

COLOR 419

yellow, and green, which it sends back. Hence, when the two
pigments are mixed, the mixture returns only those colors that
are not absorbed by either the yellow or the blue, i.e., it reflects
the greens only.

395. Complementary Colors. If we look intently at a brightly
colored blue spot for several minutes, and then turn our gaze to
a white wall, there appears to be on the wall a yellow spot of the
same sizie and shape as the blue one. In like manner, if the
spot is pink, that which appears to be on the wall will be pale
green. But we have just seen that a mixture of yellow and blue
light produces white. Hence these colors are said to be com-
plementary. This phenomenon of complementary colors may
be accounted for with the help of the Young-Helmholtz theory of
color vision. For when we look at a brightly colored spot, the
nerves sensitive to that color become tired, so that when we look
at a white surface, which is sending out all colors, the ''blue"
nerves do not respond, and the sensation corresponds to white
with the blue left out, i.e., it is that of the complementary color,
yellow

SUMMARY

1. White light is a mixture of a vast number of waves of
different lengths.

2. Difference in color corresponds to difference in wave length,
the red waves being longer than the blue.

3. The wave lengths may be measured by means of the inter-
ference fringes.

4. The velocities of different colored lights in transparent
substances, like water and glass, are different.

5. The separation of composite light into its colors is called
dispersion.

6. Dispersion makes the formation of clear images by a
single lens impossible on account of chromatic aberration.

7. Chromatic aberration may be corrected by properly com-
bining two lenses of equal and opposite dispersive powers, but
of unequal deviating powers.

420 PHYSICS

8. Every incandescent vapor sends out waves of definite
lengths which correspond to a few simple colors that are charac-
teristic of it.

9. Substances may be analyzed by the spectroscope. This
analysis may be extended to include the sun and stars.

10. Dark lines in the spectrum are due to the absorption of
definite waves by metallic vapors.

11. The atmospheres of the sun and other celestial bodies consist
of the simple substances known on the earth.

12. The colors of ordinary objects are very complex. They
absorb large numbers of waves of certain .lengths and reflect
waves of certain other lengths, which give them their character-
istic tints.

13. There is no simple numerical relation between the vibra-
tion numbers of the components of a complex color, and so we
have no color scale and no physical basis for judging concerning
harmony and discord of color.

14. The eye seems to have three sets of nerve fibers, sensitive
respectively to red, green, and violet blue.

15. The ear analyzes a complex tone and hears the elements
separately. The eye receives one sensation as the resultant of
the action of a complex color.

16. Two colors that produce the sensation of white when they
are mixed together are said to be complementary.

17. The results of mixing colored lights are quite different
from those of mixing colored pigments.

QUESTIONS

1. How may we prove that white light consists of a Idrge number of
colors?

2. How may we prove that different colors correspond to waves of
different lengths?

3. What is meant by dispersion? How does a prism deviate a
beam of light? How does it separate it into colors?

4. What is chromatic aberration? How is it corrected in lenses?

5. How is dispersion used for chemical analysis? What improve-
ments do we have to make in Newton's apparatus in order to produce
a clear spectrum?

6. Are the colors emitted by incandescent vapors simple?

COLOR 421

7. What is the difference between the spectrum of an incandescent
solid and that of a vapor? How does this difference depend on the
temperature?

8. How does resonance enable us to explain the dark lines in the
solar spectrum?

9. How is the sun's atmosphere analyzed?

10. How is the complexity of ordinary colors different from that of
a musical tone?

11. Why do we have no color scale to correspond to the musical
scale?

12. How does the eye detect differences in colors? Has it, like
the ear, a separate nerve for each separate number of vibrations?

PROBLEMS

1. When salt is burned in alcohol, the light emitted by the flame is
the characteristic yellow light of sodium. Why do people appear
ghastly in this light? If a yellow gas flame, which has a similar ef-
fect, is surrounded with a red glass globe, does it make .the people
appear more natural? Why?

2. If a piece of blue paper is illuminated with red light, what color
does the paper appear to have? What color does it appear to have
when illmninated with yellow light obtained by passing daylight
through yellow glass or a solution of bichromate of potash? If the
yellow light were that of sodium, what would be the result?

3. When you look through a prism at a window or a broad band
of white paper, you see a broad patch of white with red, orange and
yellow on one edge, green, blue and violet on the other; but if you
look at a narrow slit through which white light is coming or at a
narrow band of white paper, you see a continuous spectrum. Explain
the cause of the difference.

4. Why is it that the complementary of any one of the spectrum
colors is a complex tint and not a pure color?

5. Why do interference fringes in white light give complex tints
instead of pure spectrum colors?

SUGGESTIONS TO STUDENTS

1. Make a soap solution such as you use for blowing soap bub*
bles, and form a soap suds film over the open end of an ordinary
drinking glass with straight sides, not tapering. Set the glass on its
side so that the film stands vertically. See if interference bands ap-
pear as in the air wedge mentioned in Art. 378. If the bands do not
appear, the solution is too strong, if the film breaks, the solution is
too weak. Does a black spot appear at the top of the film? Can you

422 PHYSICS

find out how thick the film is at the bottom? Remember that the
thickness of the fihn increases half a wave length for every black band.
If you view the film in red light, and the wave length of the red
light in air is 0.000065 cm, how long is it in water, i.e., in the soap
solution if the index of refraction is 1.33?

2. Can you find out how the rainbow is formed? How does the
light pass into and out of each drop of water so as to be separated
into its colors? Why is the bow always curved? Has the' rainbow
an end? Why do we sometimes see two bows?

3. Consult a book on botany and see what you can find out about
chlorophyll and its relation to the green color of leaves. What rays
does it absorb, and which does it reflect?

4. You can buy a small prism from an optician for about 30 cents,
and with it make a number of interesting and instructive experi-
ments. See Twiss' Laboratory Exercises in Physics (Macmillan, N. Y.)
Exercise 43. This will suggest others; e.g., examine with the prism,
as there directed, narrow strips of the Milton Bradley colored papers,
which may be obtained at your bookstore, and the lights from red and
green fire, which you can buy at your druggist's.

5. The Milton Bradley color top, which can be bought postage
prepaid for 6 cents, is an endless source of amusement and instruction.
By all means get one and use it as directed in the Bradley color book,
Hopkins' Experimental Science, or Mayer and Barnard's Light.

You will find the following books of interest in connection with
the subjects discussed in this chapter: O. N. Rood, Modem Chromat-
ics (Appleton, N. Y.); N. Lockyer, Spectrum Analysis (Appleton, N.Y.) ;
Vanderpoel, Color Problems (Longmans, N. Y.); Elementary Color {MilUyn
Bradley Co., Springfield, Mass.); Mayer and Barnard, Light (Appleton,
N. Y.)- This is full of beautiful home experiments. Thompson's
Light, Visible and Invisible, Lommel's Nature of Light (Appleton, N.
Y.), and Professor D. P. Todd's New Astronomy (Am. Book Co., Cin-
cinnati) , also contain much that is interesting and not difi^icult to read.

CHAPTER XXII
VELOCITY OF LIGHT

396. The Medium. In the preceding chapters we have
assumed that light is a wave motion. We have learned how, by
ordinarily traveling in straight lines, it enables us to judge the
directions of the objects. We have then discovered some of the
consequences of considering that differences in color correspond
to differences in wave length. We now pass to the problem
of determining more exactly the nature of the light waves. If
light is a wave motion, in what medium does it travel, for we
have learned that a medium is necessary for the propagation of
w^avfes? It is not air, because light comes from the filament of
an incandescent lamp even though the air is entirely pumped
out of the bulb, as was noted in Chapter VIII. Also, light comes
from the sun and stars to us, although we are certain that our
atmosphere does not extend that far. But how shall we find out
more about the medium in which light waves are propagated?
How determine its properties? One possible way is by determin-
ing the velocity of light, as we determined the properties of air
from a study of the velocity of sound in it (c/. Art. 313). But how
shall we measure the velocity of light? It is well known that its
velocity is very great; for, as was noted when discussing the velocity
of sound, light appears to travel short distances instantaneously.

397. Galileo's Method of Measuring the Velocity of Light.
The first to propose a method of measuring the velocity of
light was Galileo. He suggested stationing two observers on
distant hills and supplying each with a lantern, fitted with a shutter
that could be closed and opened quickly. Then observer 1 opens
his lantern; when observer 2 perceives the light, he opens his.
Then observer 1 has but to note the time thr^t elapses between
opening his lantern and seeing the flash of t^ e other lantern,

. 423

424

PHYSICS

and this time should be that taken by the light to pass from
observer 1 to observer 2 and back. So reasoned Galileo. The
experiment was tried, but without satisfactory results, for it was
found that the time was too short to measure accurately. Further,
the method is inaccurate, because there is introduced the time
taken by observer 2 in becoming conscious of the appearance of
the light from observer 1 and opening his lantern. This time is
greater than that taken by the light in traveling the entire dis-
tance. The experiment is correct in principle, but the application
of the principle can be improved.

398. Fizeau's Method. The first improvement consists in
replacing .observer 2 by a mirror. Since the action of the mirror
is automatic, this change eliminates the inaccuracy due to the
operations of observer 2. But even then it is found that observer
1 sees the light before he can get his lantern fully open, i.e., he

is too slow for the light.
Therefore we have to
replace the shutter on
his lantern by a
mechanical device that
will open and shut
the lantern more
quickly. This was
done by Fizeau in
1847. The principle
of his experiment is illustrated in Fig 232. Light from the source
S passes through the hole h in the box and thence to the distant
mirror M. It then retraces its path and is partly reflected at
the glass plate G to an observer at E. On one end of the box is
a toothed wheel W, which is so placed that when it revolves the
light is cut off whenever a tooth t of the wheel is in the path of
the light, and allowed to pass whenever a notch n is in that
position. As the wheel rotates, the light is alternately cut off
and let through. By rotating the wheel rapidly we are able to
make these openings and closings of the lantern follow one
another as rapidly as desired.

Fig. 232. Pkinciple of Fizeau's Experiment

VELOCITY OF LIGHT 425

But it is still necessary to be able to measure the time accu-
rately. This may be done by the notched wheel itself, for if
we arrange matters so that the observer sees through the notches
of the wheel the light reflected from the distant mirror, the re-
turning light will be cut off if the time taken by it in traveling
from the wheel to the distant mirror and back is the same as that
taken by the wheel in turning, so that a tooth takes the place pre-
viously occupied by a notch. Now, it is easy to measure the
angular velocity of the wheel, and also to count the number of
notches in its rim; we may, therefore, determine accurately the
time taken for a tooth to replace a notch. Twice the distance of
the mirror divided by this time will then be the velocity of light.

The experiment has been made many times in this and in
other ways, and the result is that light has been found to travel at
the rate of 186,000 ^^ = 3 x 10'* ^. Since the circumference
of the earth is about 25,000 miles, it appears that light is able
to traverse in one second a distance equal to about 7 times the
circumference of the earth.

Though this seems a surprisingly great velocity, it is not so
tremendous when we consider the distances from the earth to
the sun and the stars. Thus, it has been found that it takes
about 8 minutes for light, traveling at this rate, to come from the
sun to the earth; and the distance of the nearest star is so great
that it takes light over three years to travel from it to the earth.
Further, the most distant stars that we can see are so far away
that it takes light some 5,000 years to come to us from them, and
probably there are stars still farther away. These figures show
us that the ratio of the circumference of the earth to the distance
of a faint star is that of \ sec to 5,000 years, or TTiJTTrTi^iiirTri^TrTr-
Thus the knowledge of the velocity of light helps us to form some
idea of the immensity of space and of the relative size and impor-
tance of the earth in comparison with the universe about it.

399. The Ether and Its Properties. But what of a medium
that can propagate waves at the rate of 3 X 10'° ^? Can we
apply equation (15) Art. 296, which gives the relation between
velocity, elasticity, and density? If we can, it is evident that the

426 PHYSICS

elasticity must be enormously great and the density exceedingly
small in order that the square root of the quotient may be so
large a number. But how can we measure the elasticity of this
medium? In the case of air the elasticity is easily measured
by compressing the air and measuring its change in volume.
Can we apply pressure to the medium in which light travels?
Can we measure its density? These quantities are evidently so
small that we can not measure them by mechanical means; for
when we have pumped, the air out of an incandescent lamp bulb
as far as is possible, our gauges indicate no measurable pressure,
and we have not been able to weigh or measure the medium in
* finy mechanical way. But though we are not able to determine
the numerical value of these factors for the medium, we can give
it a name. So we call it the ether. We shall have occasion to
learn more of the properties of the ether as we proceed. At
present we can only conclude that whatever properties it may
have, it does not react in any measurable way to the ordinary
mechanical forces, such as pressure, torsion, and the like.

Since we thus learn that we can not measure the properties
of ether as we can those of air by mechanical means, we are com-
pelled to seek elsewhere for some method of finding out what its
nature is. Are there not other phenomena which involve the ether,
and by means of which we may study its character? We have
already noted, in the study of electricity, that electricity and mag-
netism do not act by means of air, but through some other me-
dium (Art. 286). This fact has led scientists to investigate
carefully whether those properties of the ether which may be
discovered by means of experiments in electricity and magnetism
can in any way assist us in framing a theory concerning the prop-
erties of light waves.

400. Electric Waves. When a Leyden jar is discharged,
the electric discharge vibrates very rapidly back and forth a
number of times between the terminals (cf. Art. 197). There-
fore we have in the electric spark a vibrating something which
may send out waves. Can we detect at a distant point the
discharge of a Leyden jar in any other way than by the sound

VELOCITY OF LIGHT

427

Fig. 233. Electrical Resonance

and light of the spark? If we place a second jar, in every way
like the first, in close proximity to the first, we notice that a spark

passes between its

terminals at S, Fig.
233, whenever one
passes between the
terminals of the first
jar. We must then
conclude that the
second jar has electric
oscillations generated
in it by resonance.
But resonance implies
waves; and so we con-
clude that the spark, probably, is a source of electric waves.
These waves, as is now well known, can also be detected in other
ways; in fact, they are the waves with which wireless telegraphy
operates, for wireless messages are sent by discharging electric
sparks. So we learn that electric sparks generate electric waves
which travel to distant points and may there be detected.

If electricity acts by means of the ether, these electric waves
must be ether waves. Therefore it is of interest to find out how
fast they travel, because this knowledge will enable us to com-
pare them with light waves. We can measure their velocity by
means of the equation v = nl (Art. 296), since the number of
vibrations can be calculated from the dimensions of the jar used
in sending the sparks; and the wave length can be measured by
causing the impulses to form stationary waves on long wires, and
measuring the distance between the nodes (Art. 301). The
first experiments of this nature were performed by Hertz in 1888,
and his result ushered in a new epoch in physical science, for he
found that these electric waves travel with the same velocity as
light does.

The conclusion which we draw from this remarkable result
is that the ethers of light and of electricity are, the same; and,
further, that light is an electric vibration, not an elastic one.
Since the announcement of this admirable theory, many other

4^8

PHYSICS

facts have been discovered that all add weight to the argument,
and make us more content with the simplicity and the elegance
of the conclusion.

401. WireleiB Telegraphy. When these electric waves are
used in transmitting messages without wires, they are started by
the spark from an ordinary induction coil. The receiving appa-
ratus (Fig. 234) is very interesting. It consists of a small glass
tube C, to which are fitted two metal plugs j> p. The ends of
these plugs are about 1 mm apart, and the space between them

Fig. 234. Diagram of receiving station for wireless messages

is filled with loose nickel or silver filings. A current from a small
battery b passes through this tube^and the relay R, and back to the
battery. When the circuit is thus completed, very little current
flows, because the resistance of the loose filings is large, and
therefore the relay armature is not moved. But when electric
waves fall on the filings, their resistance is in some way diminished.
The current through the filings and the relay is thus increased,
and the relay armature is pulled toward the magnet. This action
of the filings is called coherence, and the tube C is called a co-
herer. When coherence takes place, the relay closes a local
circuit in which a sounder S is operated by a battery B, as in
the ordinary telegraph system.

In order to break the circuit through the coherer, it must be
tapped so that the filings are shaken up. It is, therefore, usually
placed so that the armature of the sounder strikes the coherer

VELOCITY OF LIGHT 429

when it flies back. The sensitiveness of the apparatus may be
much increased by connecting one end of the coherer to earth and
the other to a long wire stretched vertically on a high pole. Such
a vertical wire is called an antenna. In Uke manner, the power of
the sending coil may be increased by attaching one of its termi-
nals to an antenna and the other to earth. A Leyden jar across
the tei-minals of the coil often helps. The points at which the
coherer and the sounder circuits are broken must be shunted
with coils of fine wire of high resistance, for if a spark occurs when
a circuit is broken, it sends out waves that affect the sounder and
confuse the signals.

402. The Complete Spectrum. We are now in a position to
enlarge our ideas of the spectrum. Since the electric waves travel
with the velocity of light, and since their numbers of vibrations
vary from about 1000 to 6 X 10^^ per second (c/. Art. 197), their
wave lengths vary from 3 X 10^ to 0.5 cm. In Art. 397 we learned
that a long red wave has a wave length of 0.000066 cm and a short
blue one a length of 0.00004 cm; we now find that the electrical
vibrations send out waves of greater length than those correspond-
ing to red light. We have also noticed that as a body is heated,
it sends out first longer heat waves, then shorter heat waves, and
finally, at a temperature of about 520*^ C, it begins to emit red
light waves. Putting these three facts together, we may conclude
that the entire spectrum is much longer than appears to the eye,
for it must contain heat and electrical waves beyond the red.
This portion of the spectrum is therefore called the ultra-red,
or heat spectrum. Investigation shows that this spectrum really
exists; for if we place in the solar spectrum beyond the red an
instrument capable of detecting heat — for example, a thermometer
— we shall find that it indicates heat action there.

Further, if we take a photograph of the visible spectrum,
we find that the photograph indicates the presence of waves
beyond the violet end; i.e., we can photograph more than we can
see. So we conclude that there are electric waves shorter than
those corresponding to violet light, and that these waves can act
on a photographic plate. This extension of the spectrum is called

430 PHYSICS

the ULTRA-VIOLET, OF photo-chemical spectrum. The shortest
of these photo-chemical waves that has been measured, has a wave
length of .00001 cm.

We thus see that the entire spectrum contains a large number
of waves in addition to those that produce the sensation of
sight, for it contains waves varying in length from 3 X 10^
cm to those of length .00001 cm. Of these waves, those lying
between the limits 3 X 10^ and .5 cm are called electric waves,
and are able to act on electrical apparatus; those between the
limits .008 and .00008 manifest heat action, while those between
the limits .00008 and .00004 affect the eye and are called light.
Those that are shorter than .00004 cm may be detected by their
action on a photographic plate. It is possible that waves exist
beyond these limits, but they have not been detected by any
of the devices at present known to science. It is the present belief
of scientists that all these waves are of the same nature, i.e., that
they are all electric waves.

403. What Spectra Tell Us. The detailed study of spectra
is of great importance in physics, not only because we can thereby
analyse chemical compounds, but also because it opens a pos-
sible way of discovering the nature of the vibrating particles
which are the sources of light waves. The principles on which
this study is based have been discussed in what has preceded,
but it may be well to bring them together here for the sake of
showing their relations. Thus, we have seen (Art. 340) how a
vibrating string sends, out waves whose lengths are related by the
simple ratios, 1, i, J, etc. Conversely, when we receive such a com-
plex sound wave, and on analysis fin(J that its component vibra-
tions are related by these simple ratios, we are justified in con-
cluding that the source of the wave is either a vibrating string, a
rod, or some other body which is capable of sending out that
particular series of overtones. Now, all bodies that send out
such a series of overtones are long and thin, i.e., they resemble a
straight line in geometrical form. So when we have analyzed a
complex wave and found it to consist of a series of simple waves,
whose lengths are related by the ratios, 1, J, J, etc., we conclude

VELOCITY OF LIGHT 431

that the geometrical form of the vibrating source of the waves is
a straight line.

If, however, the complex wave is found, on analysis, to con-
tain the overtones characteristic of a bell, we are justified in con-
cluding that the vibrating body is shaped like a bell; and, simi-
larly, if the component vibrations are those characteristic of a
solid ring, or of a sphere, or of a body of any other particular
geometrical form, we infer that the source of the waves has that
particular form. A similar conclusion may be drawn concerning
the geometrical form of a vibrating source of light; for we have
learned how each incandescent vapor sends out a certain series of
simple vibrations characteristic of it. We can sort out these
simple vibrations by the spectroscope, determine their vibration
numbers, and so find out whether any numerical relation exists
between the vibration numbers that correspond to the bright lines
in the spectrum of any element. Careful experiments have
proved that such numerical relations do exist among the vibration
numbers that correspond to the bright lines of many of the chem-
ical elements. What, then, is the geometrical form of the body
that can vibrate in such a way as to send out the corresponding
vibrations? What is the geometrical form of the vibrating body,
i.e., of the atom?

To this question science has not yet found an answer; for
though we know the relations among the vibration numbers of
the series of waves sent out by the atoms of many substances,
that relation is very complex, so that we have not yet been able to
show what geometrical form is capable of vibrating in such a way
as to send out that series of vibration^. The problem, however, is
not hopeless; and during the last few years considerable progress
has been made towards its solution. All that can be said at
present is, that this study of spectra offers a possible way of en-
abling us to form some conception of the shape and construction
of atoms.

The importance and wonders of spectra thus become clear;
for we see that from them we can not only discover the make-up
of stars at almost infinite distances from us, but we can also study
the mechanism of the tiniest thing that the human mind has been

432 PHYSICS

able to conceive. We see that we are surrounded by phe-
nomena of marvelous complexity, yet governed by simple prin-
ciples — how these phenomena and the principles that govern
them extend to both the infinitely great and the infinitely small,
without any conceivable limit. For who can set an outside bound-
ary to the universe, or who can measure the size of the smallest
particle which plays its part and has its own use in the
economy of Nature? Yet throughout this vast complexity of
relations and interrelations, among infinitely small atoms and
the infinitely great universe, we can discern the operation of com-
paratively simple principles, which are manifest in the least, as
weir as in the greatest, of the operations of the universe.

SUMMARY

1. The velocity of light is 3 X 10^"^.

2. The elastic constants of the ether can not be measured by
mechanical means.

3. The electric spark generates electric waves varying in
length from 3 X 10^ to 0.5 cm.

4. These long electric waves travel with the velocity of light.

5. Heat and light waves are of the same nature as electric
waves, and the ether is the medium which transmits them all.

6. The complete spectrum contains electric, heat, light and
photo-chemical waves.

7. The spectrum enables us to study the geometrical form of
atoms as well as the constitution of the heavenly bodies.

QUESTIONS

1. How is it possible to measure the velocity of light?

2. How long does it take light to travel a distance equal to the
circumference of the earth? How long to come from the sun? From
a star?

3. What medium transmits light? Can we measure its elasticity
by mechanical means?

4. What other phenomena besides light depend on the ether for
their manifestations?

5. How are electric waves generated? How detected? Describe
a wireless telegraph system and explain its operation.

VELOCITY OF LIGHT 433

6. What is the velocity of propagation of electric waves? How is
their velocity measured?

7. Why do we conclude that the waves of heat and light are electric
in their nature?

8. What different kinds of waves make up the complete spectrum ?

9. Can we study the geometry of an atom by the spectrum? How?

10. What are the limits of the domain within which the study of
the spectrum is valuable?

PROBLEMS

1. A certain Leyden jar is discharged and the oscillations of the
spark are found to have a period of ToioTV s^c; what is the length of
the electric wave started by the spark? If the wave length of so-
dium light is 0.000059 cm, how many oscillations does a sodium par-
ticle execute per second?

2. Since incandescent bodies are radiating long heat waves as well
as light waves, much of the energy supplied to them is wasted, as far
as light making is concerned. For example, the unit of light is the
candle, and it has been found that a unit candle radiates light energy
at the rate of about 19 X 10' ^^. At what rate does a 16 candle in-
candescent lamp radiate light energy? Such a lamp requires for its
maintenance to be supplied with energy at the rate of about* 55
watts; what is its efficiency as a light producer?

3. In Art. 170 the heat of combustion of illuminating gas was given
as 18 X 10* gm cal per cubic foot. An ordinary gas flame consumes
5 j- — and radiates with an intensity of about 16 candles. What is the
efficiency of the flame sis a light producer?

4. An arc lamp radiates with an intensity equal to that of 2000
candles and requires an expenditure of about 500 watts, what is
its efficiency?

5. Is it more economical to bum illuminating gas in a gas en-
gine, and use the engine to run a dynamo, and let the dynamo
feed an arc lamp, than it is to bum the gas directly for its light?

6. A cm? of the earth's surface, perpendicular to the path of

the sun's rays, receives light energy from the sun at the rate of

7 X 10^ ®— ^. A cm^ distant 1 m from a standard candle, and per-

sec ^ ^

pendicular to the rays, receives light energy from the candle at

the rate of 15 ^^. How many candles at the distance of 1 m

sec
would be necessary to give the same intensity of illumination per

cra2 as is given by the sun? If the distance to the sun is 15 X
W^ m, how many candles at the distance of the sun would be re-
quired to illuminate the earth with the same intensity as the sun
does?

434 PHYSICS

SUGGESTIONS TO STUDENTS

1 . If there is a wireless telegraph station near you, visit it and
find out how it works. Ask the operator if they are able to tune
the receiving instrument so that it will respond only to messages
intended for it.

2. What can you find out about the efficiency of a fire-fly?
How do they produce so much light with so little expenditure of
energy? May the phosphorescence of decaying wood be a similar
phenomenon?

3. You will find interesting reading on the topics of this chap-
ter in Oliver J. Lodge, Signalling Through Space Without Wires (Van
Nostrand, N. Y.); R. T. Glazebrook, J. Gierke Maxivell and Modern
Physics; Lodge, Modern Views of Electricity (Macmillan, N. Y.); and
Thompson's Light, Visible and Invisible.

4. It is not a very difficult task, with the help of Profe^ssor Lodge's
book, to make a coherer and connect it as in Fig. 234, so as to re-
ceive wireless signals from the sparks of a small induction coil or
electrostatic machine. Many boys have made their own outfits, in-
cluding the induction coil, and operated them successfully over con-
siderable distances.

CHAPTER XXIII
ELECTRONS

404. Origin of Light Waves. In this final chapter we shall
endeavor to give an outline of the argument on which our present
theory concerning the nature of the ultimate particles of matter
is based. What has preceded has made us familiar with the
main facts on which this argument is founded.

The first point in the theory is the hypothesis that light con-
sists of ether waves, and from this we must infer that the source
of those waves is a vibrating something. Further, since these
waves are so minute, the vibrating something must be very small,
so as to be able to vibrate very rapidly — some 6 X 10'* times
a second. We are thus led to conceive that the sources of light
waves are minute vibrating particles.

405. Electrons. Since the waves of light are waves in ether,
ih^ vibrating something must be of such a nature that it is able,
by its vibrations, to disturb the ether, and so become a source of
waves. Therefore, in order to conceive how these minute vibrating
particles can disturb the ether, we are led to the further hypothesis,
that each of them carries an electric charge; for we have learned
that a vibrating electric charge, such as that with which we
have become familiar in the sparks from a Leyden jar, can start
electric waves similar to the light waves (Art. 402). The hy-
pothesis that there are in Nature tiny particles which carry elec-
tric charges is further justified by the phenomena of electroly-
sis; for we have learned (Art. 286) that the actions of the ions
in an electrolyte is described most simply by assuming that they
carry electric charges. So we have come to believe that light
waves have their origin in the vibration of minute electrically
charged particles. These particles have been named electrons.

435

436 PHYSICS

Do they manifest themselves in any other way than as sources
of Hght waves? Are they the same as the ions of electrolysis?
How large are they? How are they set into vibration?

406. Crookes' Vacuum Tubes. Taking up these questions in
order, we may answer to the first that we have good reason to
believe that the phenomena observed in connection with electric
discharge in vacuum tubes are due to these electrons. What are
some of these phenomena? Let us analyze the action in a vacuum
tube as the air pressure is gradually diminished. Thus, if we
send the spark from an induction coil through one of these tubes,
the discharge inside the tube, if none of the air has been pumped
out of it, will, of course, be exactly like that in air. But if we
connect the tube with an air pump, and begin to pump the air
out, the appearance of the discharge changes. It first becomes
less like the familiar, sharply defined, zigzag flash. Fig. Ill, and
spreads out in a broad band, giving a diffused purple glow.
Presently, as more air is pumped out, this light appears to grow
whiter and fill the tube. As the exhaustion is continued, the
whiteness disappears, and we notice a pale blue streamer extend-

Jng from the negative electrode and perpendicular to its surface.
Since this streamer proceeds from the negative pole or cathode,
it is called the cathode beam and is said to consist of cathode

RAYS.

407. Cathode Bays. These rays have been carefully studied
and found to possess many peculiar properties. We find, in the
first place, that where they fall on the glass walls of the tube or on
certain substances placed in their path, they cause these substances
to emit light. They are then said to produce fluorescence.
In the next place, the cathode rays travel in straight lines per-
pendicular to the surface of the cathode, for such fluorescent light
always appears in the direction perpendicular to the surface of
the cathode, and the shadow cast by a metal screen is perfectly
well defined, as shown in Fig. 235. Finally, they are sensitive
to a magnetic field; for if a magnet is brought near the tube, the

ELECTRONS 437

positions of the fluorescent spots change. In the light of these

facts, what assumption may we make as to the nature of these

rays? All the phenomena •

exhibited by them may be

simply described if we

assume that they consist of

minute particles, carrying

electric charges, and shot

out from the cathode by the

electric force there acting.

Since the cathode is the

negative pole, it is clear fiq. 235. Shadow in Cathode Rays

that the charges carried

away from it by the particles must be negative.

It is easy to see how such negatively charged particles, when
shot from the cathode, would be electrostatically repelled along
straight lines perpendicular to the surface of the cathode. It is
also easy to comprehend how they can produce fluorescence when
they strike, by jarring the smallest particles in the substance
on which they strike, and so causing them to shiver and send out
light. But why should the cathode rays be bent from their straight
path by a magnet? To understand this, we must recall the fact
that a charged particle in motion produces a magnetic field (Art.
227). Therefore, if the cathode rays consist of charged particles
traveling in straight lines, they should produce a magnetic field;
and if they do so, of course, the cathode beam itself would be
magnetic and would be repelled or. attracted by an outside
magnetic field.

We can thus understand how the phenomena exhibited by the
cathode rays are simply described by assuming that those rays
consist of negatively charged particles shot out from the cathode
and traveling in straight lines. So we have found that in the
phenomena of the vacuum tube we may be dealing with electrically
charged particles or electrons. We may then ask whether these
cathode particles may not be similar to those whose vibrations
we have conceived to be the source of light waves. We can best
arrive at a conclusion on this matter if we first find out something

438 PHYSICS

concerning the size of the electrons in the cathode rays. But
can we determine their size?

408. Size of Electrons. Science has been unable to answer
this question until very recently, for it is no easy matter to de-
terinine the mass of particles so small that the weight of millions
of them together could not be detected by our most sensitive balance.
However, the bending of the cathode rays by a magnetic field
enables us to approach a solution in the following way: If the
cathode rays consist of particles each carrying an electric charge
e and traveling with a velocity v, it is clear that the strength of
the magnetic field generated by them will be proportional to
both e and v (c/. Art. 227). Therefore, the force acting between
the field of each particle and that of the external magnet, //,
will be proportional to e v II (Art. 205). But this force gives the
particle a sideways acceleration, and this is clearly proportional
to the magnetic force e v H, and also inversely proportional to
the mass m of the particle. Therefore the sideways acceleration

is proportional to . Now, it is easy to measure //, and it

has been possible to measure v\ and since we can also measure

the sidewavs acceleration, we can determine the value of — ; i.e.,

of the ratio of the charge of a particle to its mass. This ratio
is found to have the numerical value of 1.87 X 10^. The value
of this ratio is the same no matter of what substance the cathode
is made.

Now, we have conceived (Art. 227) that the particles acting in
electrolysis are charged particles, and that they move under the
action of the electric forces in the solution. Evidently the velocity
of their motion will be proportional to the charge e which each
carries, to the strength E of the electrostatic field, and inversely
proportional to the mass m of the particle; i.e., their acceleration

will be proportional to — . It is easy to measure their accelera-
tion, and also to determine E, and so we reach another value of
this ratio — ; but this time it is for hydrogen 10*.

ELECTRONS 439

What is the reason for this difference in the two results? Are
the charges e the same, and the mass of the cathode particles
less, or are the masses the same, and the charges of the cathode
particles greater? The answer to this question we owe to the
skill and ingenuity of J. J. Thomson, of Cambridge, England;
for he has succeeded in showing by experiments, whose descrip-
tion would lead us too far from our present argument, that the
charges in the two cases are the same, and therefore that the
masses of the cathode particles are much smaller than those of the
ions with which we deal in electrolysis. But in electrolysis we
have every reason to believe that the smallest masses of the ele-
ments involved, i.e., the ions, are the atoms. Therefore we con-
clude that the cathode particles or electrons are much smaller
than atoms.

The relative sizes of the atom and the electron may be ob-

tained by dividing one value of the ratio — by the other. Thom-
son thus arrived at the conclusion that electrons are so small thai
it takes 1870 of them to make a hydrogen atom. Experiments with

other elements than hydrogen tend to show that the ratio — in

electrolysis is inversely proportional to the masses of the atoms,
and therefore we must conclude that the number of electrons
necessary to produce a mass equal to that of an atom of any ele-
ment is 1870 times the mass of an atom of that element. This
conclusion has recently been confirmed by other experiments
along wholly different lines; so we now believe that an atom is
composite and contains many electrons in its make-up.

Thus, at last, science seems to have found in the electron a
particle which is smaller than the atom and which may be found
to be the ultimate particle of matter.

409. Eadio-Activity. But are electrons found free in nature,
or do we know them only in vacuum tubes? The study of the
recently discovered phenomena of radio-activity leads us to believe
that the radium rays consist of the emission, by radium and other
similar svbstances, of electrons carrying negative charges; for the

440 PHYSICS

emanations of these substances act in many ways like the cathode
rays. Probably many of you have seen one of Professor Crookes*
spinthariscopes, in which a small particle of radium is mounted
over, a screen of some substance, like zinc sulphate, which becomes
fluorescent when cathode rays fall on it. On observing the screen
in a dark room, it is seen to be scintillating all over with tiny sparks.
These are believed to be produced by the bombardment of elec-
trically charged particles that are shot out by the radium so as
to strike the fluorescent screen. This phenomenon and many
others lead us to believe that electrons exist free in nature, since
they are emitted by the radio-active substances.

410. How Electrons Start Ether Waves. We may now re-
turn to the phenomena of light, and ask how the conception that
atoms are made up of large numbers of electrons assists us in
conceiving a mechanism to describe the origin of light waves.
We have already learned that light waves are brought into ex-
istence by sufficiently heating any substance. We must then try
to form a picture of the way in which the heat energy may be con-
verted into energy of vibration in solids and in gases. The first
point that must be noticed in this connection is that heat expands
the body; and therefore, if we conceive the body to consist of
small particles, these must be separated from one another by the
action of the heat. Further, we have noticed how all bodies are
sending out long heat waves at all temperatures (Art. 149), and
from this fact we must conclude that the small particles of which
the body consists are vibrating even at the low temperatures.
Therefore we may imagine that heating a body both increases
the amplitude of the vibrations of those small particles, and also
separates them further from one another.

When these particles form a solid, they are very close together,
and therefore, when their vibrations are rendered more intense
by heating, they must collide with one another more frequently
and with greater energy than at a lower temperature. When
two particles collide, each receives a shock, which must cause
its component parts to shiver and send out for a brief instant a
large number of waves. These waves are not those corresponding

ELECTRONS 441

to the natural periods of vibration of the particles. Such vibra-
tions are called forced vibrations (Art. 333), and they die out
very rapidly. But when the particles make up a solid substance,
the impacts between them take place so frequently that almost all
the vibrations sent out by a solid are forced vibrations. There-
fore the spectrum of a solid sending out vibrations under these
conditions, contains all possible wave lengths, and so it is a con-
tinuous spectrum. We may thus draw a mental picture of the
mechanism by which continuous spectra are formed.

But when the particles are separated by considerable dis-
tances, as they are when the substance becomes a vapor, the
collisions between the particles are less frequent, and the particles
travel a considerable distance between those impacts. Now,
although at the instant of impact they send out forced vibrations
of all sorts of periods, these quickly die away, and thus leave the
particle free between impacts to send out its own natural vibra-
tions; therefore the spectrum of incandescent vapors consists
mainly of vibrations that are characteristic of tJie particles them-
selves, for each has its own particular natural period and has
plenty of room in which to vibrate.

411. X-Eays. Another interesting form of radiation is that
found in the X-rays. The apparatus used for generating these
rays is shown in Fig. 236. So much has been written in the
magazines about these rays, and their applications to surgery
have brought them so prominently before the public, that we
need only mention some of their most marked characteristics.
These are: 1. They appear to originate wherever cathode rays
fall on matter of any kind. 2. They travel in straight lines, but,
unlike light, they are not reflected or refracted. Thus they can
not be deviated by a prism or brought to a focus by a lens. 3.
Unlike the cathode rays, they are not deflected by a magnet.
4. The depth to which they penetrate material substances is
nearly proportional to the densities of the substanc»es. Thus
they pass readily through wood and animal tissues, which are
opaque to light. 5. They produce fluorescence in some sub-
stances, like platinum-barium cyanide and calcium tungstate.

442

PHYSICS

These substances are therefore used in making the screens
with which shadows of the bones may be seen. 6. They act on

Fig. 236. A Modern X-Ray Coil and Tube

a photographic plate, so that shadow pictures of the bones may
be made. Fig. 237 is such a picture of a hip joint. 7. When
they fall on an electrically charged body, the charge quickly
escapes.

What are these rays? Since they are not refracted or re-
flected as light is, they probably do not consist of waves. Further,
they are not deflected by a magnetic field, and therefore they
probably are not projected charged particles. In order to gain
some conception as to what they may be, consider the way in
which they are generated. A negatively charged particle, travel-
ing with a great velocity, strikes on some sort of matter. Such
a traveling electron produces, while in motion, an electric current;
but when its motion is suddenly stopped the current it is producing
stops also, and this sudden stopping of the current must give the
ether a quick jar or impulse. This impulse in the ether might be
likened to that produced in air when a hammer strikes a block of
wood. There is a sudden, sharp impulse of sound and we hear a
click. Such an impulse is not a wave, though it is propagated
like a wave in that it travels with the velocity of sound in air,

ELECTRONS

443

and also in that it can be heard. Similarly, we are led to the con-
ception that X-rays may be a series of such sudden impulses in
ether — an ether phenomenon analogous to the sound phenomena
produced by a Fourth of July celebration. Such a series of

Fig. 237. X-Ray Photograph

impulses would travel with the velocity of light, would not be
reflected or refracted as light is, and would not be deflected by a
magnet; i.e., it would possess the properties exhibited by a beam
of X-rays.

412. White Light. This conception of a series of impulses
in ether may help us to conceive of the nature of ordinary white
light. For we have seen how such light consists of a vast complex

444 PHYSICS

of waves, each originating at one electron in a complex atom.
We have also noted how the nature of the complex vibration
changes with every impact of the atom. Therefore we may
conceive that white light consists of a vast complex of compara-
tively short wave trains, each containing a few hundred thousand
waves. Hence a wave front can not be said to exist in white light,
since there is no plane or line in which all the particles are in the
same phase at the same time. This same conclusion holds even
for monochromatic light. Thus we find that the wave fronts of
which we have talked are only convenient mathematical fictions,
which make it possible for us to discuss in a rough sort of way
the marvelously complex phenomena of light.

413. Properties of Electrons. Before closing this discussion
it will be well to recall the facts and theories which we have been
studying, and to see if we can arrangfe them into a satisfactory
whole. To do this, let us review the main outlines of the argu-
ment, and then try to show how the results obtained by it may
lead to clear conceptions as to the mechanism of these phenomena
of Nature. 1. We have learned that it has been possible to rec-
ognize the existence of particles smaller than atoms — about ygVir
of the size of a hydrogen atom. 2. These particles have been
found to carry negative electric charges, and to travel with high
velocity. 3. It has also been possible to conceive how their
oscillations within atoms may produce the waves which we call
heat and light. 4. We have seen how these charged particles,
or electrons, seem to be of the same size and nature, no matter
from what substance they come. 5. We have learned that X-rays
are generated when these particles strike on matter. 6. We
have found that electrons are continually being shot out by radio-
active substances.

414. Positively Charged Particles. We may noW ask, Are

positively electrified particles known? Electrons are always
charged negatively. The answer to this question may be
obtained from a further study of radio-activity. It is found that
radio-active substances emit three kinds of particles. These

ELECTRONS 445

three kinds have been named, from the Greek letters, the alpha,
the beta, and the gamma particles, and they have different prop-
erties. The beta particles are found to act in many ways like
the electrons; i.e., they have a mass about y^Vu ^^ that of the hy-
drogen atom; they travel with a velocity nearly equal to that of
light, and they carry negative charges. Further, their ability to
penetrate into substances depends only on the density of the
substance, being inversely proportional to it. The alpha particles,
on the other hand, are much larger, travel with a velocity of only
tV that of light, and carry positive charges. These particles can
penetrate into substances much less easily than the beta par-
ticles. However, on account of their greater mass, they possess
greater kinetic energy than the beta particles. Their mass is
found to be about the same as that of an atom of helium; i.e.,
about that of two hydrogen atoms. The nature of the gamma
particles has not yet been determined.

415. The Theory of Atomic Structure. The phenomena of
radio-activity are believed to consist in the spontaneous break-
ing up of the atoms of the radium or of the other substances.
And since this disintegration produces both positive and negative
particles, we have to conceive that the atoms consist of both.
In fact, we should have to conceive that both exist in atoms in
order that they remain stable, for a large number of negatively
charged particles would repel one another and not stay long to-
gether in one group, unless they were held there by some posi-
tive attracting force. So we are led to believe that the atom con-
sists of a positively charged particle about which a number of
electrons are rotating, like the planets about the sun; i.e., we
imagine that the atom of matter is constructed on the same general
plan as the solar system, which may thus be considered as an atom
of the universe.

Professor J. J. Thomson, to whom we are particularly in-
debted for the experimental work on which this new theory of
matter is based, has shown how such complex particles might be
formed, and what arrangements of electrons about the central,
positively charged particle are mechanically possible. He has

446 PHYSICS

compared his results with the order of the chemical elements
according to their chemical properties as the chemists have ar-
ranged them, and finds almost entire agreement. While this fact
is of great importance and of wonderful interest, it must not be
taken to be a proof that atoms are really so formed. All that we
can prove is that it is one possible way.

Those who are interested in the remarkable and rapid prog-
ress that has been made in the last ten years by science in thus
prying into the nature of atoms will find a very good account of
the theory and its consequences in Whethan, Recent Development
of Physical Science (London, Murray, 1904). The limitations of
this, our work, make it possible to give only the barest outlines
of the subject.

416. Conclusion. We must not, however, be led to think
that science has now solved the riddle as to the nature of matter.
For, even if we have discovered the mechanism of atoms, we have
only pushed the bounds of ignorance one step further back. Though
we may be able to say that the atom is not indivisible, but is con-
structed in such and such a way, we have still to show what elec-
trons are, what the ether is, what an electric charge is, and whence
they all come. It must, nevertheless, be clear to every one who
has read this book carefully, that nature is not a vast chaos of
chance happenings, but a well ordered and governed whole.
When we study thoughtfully the phenomena about us, we must
realize that there are some simple and universal principles which
are manifest in them all. Therefore, let us leave our study with
this idea: that the universe in which we live is a marvelously
organized and governed unit. And when we try to imagine how
such a unit could have been developed, we are compelled to rec-
ognize that it could not have come to its present perfection if it
originated in an unthinkable chaos, and organized itself solely
by the interaction of blind matter and undirected motion.

INDEX

Aberration, spherical. 392; chromatic, 411

Absolute temperature, 143

Absorption, of radiant heat, 164, of light, 416:
color produced by. 416; and radiation, 166;
spectra, 414

AbeciBsa, 18

Accelerated motion, relation of d'.stances to times,
22; graphical representation, 23; laws of, 27

Acceleration, defined. 21; negat.ve. 26; determina-
tion of, 28; of gravity, 31; varies wiUi force,
33; varies with mass, 35; of gravity, same for
all bodies, 39. 323; angular. 98; towards
center, 104

Accidentals, in musical scale. 346

Achromatic lens. 410

Activity, rate of doing work, 50

Air, weight of, 115; density of, 125; pump, 126;
atmospheric pressure. 117; critical temper-
ature of, 150; liquid, 173; columns, vibrating,
335

Alcohol, boiling point of, 148; critical temperature,
150

Altematmg current transmission, 280

Aluminum, reduction of. 280

Ammeter. 268

Ammonia, critical temperature of. 150; use in
freezing. 174

Ampere, theory of magnetism. 235; unit of cur-
rent. 266

Amplitude, of waves, 302; and intensity of
sound, 337

Analysis, spectrum. 412

Angle, unit. 98; lens. 386; visual. 391

Angular; units, 98; size of image, 387

Anode, 287

Aperture, angle of, 401

Archimedes' 8 principle, 126

Arc lamp, 265; regulator for, 282

Arm of force, 75; of mass, 106

Armature, relay. 225; motor 231; dynamo. 253

Astronomical telescope, 393

Atom, theory of construction of. 430, 445

Atmosphere, pressure of, 117; water vapor m, 165;
of the sun, 415

Audition, limits of, 350

Axes, coordinate. 18
Axis, optical, 384

Bach, Johann Sebastian, 343

Back pressure in steam engine, 178

Balance, equal arm, 89

Balloons, buoyancy of, 129

Banju, 334

Barometer, mercurial, 117

Battery, voltaic, 283; storage. 291

Beats, cause discord, 350

Bell, electric 227; sounds of. 365

Bessel, gravity with pendulum, 323

Boiling point, defined, 148; of water. 148; of

alcohol, 148
Boyle, Robert, air pump. 125; law of. 131
Brushes, motor, 231; dynamo, 253
Bunsen, photometer, 399
Buoyancy, 129

C5-G-S system of units, 15

Calorimeter, 145, 275

Calorie, gram, 144; mechanical equivalent of,

171
Camera, pin-hole, 372; photographic. 391
Candle, standard, 398
Carbon dioxide, critical temperature. 150; use in

freezing, 174
Carborundum, manufacture of, 280
Cathode. 287; rays, 436
Cell, voltaic. 217. 283
CeUo, 334

Center of mass, 81; determination of, 83
Centimeter, defined, 15
Centrifugal force, 105
Centripetal force, 105

Charges, moving electric have magnetic field. 238
Charies. law of. 140
Chimes, 365

Chromatic aberration. 411
Chromic acid cell, 288
Circuits, divided, 281; electric, 284; magnetia

215
Circular motion. 103
Clarinet, 336

447

448

PHYSICS

Climate, cITcct of water on. 153

Coefficient of expansion, of Rases, 140; of solids,
143; of liquids. 143: linear. 143

Coherer. 428

Cold storage, 174

Collecting rings. 231

Color, and wave length, 370; complex, 415; com-
plementary, 419

Columbus, Christopher, magnetic declination, 215

Commutator, motor, 232; dynamo, 253

Compass, mariner's, 211; declination of, 215

Complementary color, 419

Complex waves, 308; tones, 360; color, 415

Component motion, 67

Composite machine, mechanical advantage of, 85

Composition of motions, 57, 67; of forces, 63

Compound microscope, 397

Compressed air, for drills, 112, 122

Concave, mirror, 383; lens, 395

Condensers, steam engine, 179; electric, 201

Condensing pump, 113

Conduction, of heat, 158; electrical, 192

Conductors', defined, 192; charge resides on out-
side of, 200

Conjugate foci, 388

Conservation of energy, law of, 44, 237

Convection, of heat, 158

Convex lens, 376

Cooling, by expansion of gas, 172; by evaporation,
173

Coordinate axes, 18

Copper, in voltaic cell, 217, 287

Comet, 336

Corti, fibers of, 350

Coulomb, law of electrostatic force, 200

Counterpoise on driving wheels, 84

Crane, traveling, 61

Crests, of waves, 301

Critical temperature, 150

Crookes, Sir William, vacuum tube, 436

Ctesibius, 114

Currents, electric, 216; possess magnetic field. 219;
production by cells, 217; unit, 266; induced
by a magnet, 245; induced by currents, 248;
laws of, induced, 250; heating effect, 275

Dark line spectra, 414

Davy, Humphrey, arc lamp, 265; electrolysis, 289

Declination, magnetic, 215

Density, defined, 41; of water equals 1. 41; de-
termination of, by Archimedes' 8 principle, 129
of air, 125; and velocity of waves, 306

Deviation, minimum, 410

Dew, 152

Diar)hragm, telephone 250; lens, 302

Dielectric, 201; strain in, 203

Diffuse reflection, 379

Diffusion, 160

Direction, jxirccption of, 371

Discord. 352, 364

Dispersion, 407

Displacement and Work, 42; electric, 203; pro-
portional to force in simple harmonic motion,
319

Disruptive discharge, oscillations of, 203; starts
electric waves. 427

Distance, unit of, 15

Distillation, 157

Distinct vision, limit of, 391

Dollond, John, achromatic lens, 412

Dominant triad, 343

Dufay, law of, 194

Dyes. 418

Dynamo, principle of, 251; machine. 252; alter-
nating current, 255; efiSciency of, 271

Dyne, unit force, 38

Ear, 350; perception of complex tones by, 364
Earth, magnetism of, 212, 215; rotation proved by

pendulum, 324
Efficiency, 87; of locomotive, 180; depends on

absolute temperature, 180; of triple expansion

engine, 182; of gas engine, 183; of the dynamo,

271
Elasticity, of fluids, 118; of air, 330; and velocity

of waves, 306
Electric charges, positive and negative, 194; at-
traction and repulsion of, 194; generated by
* separating unlike bodies, 196; imit, 201
Electrification, two kinds, 193
Electric motor, 234; series, shunt, and compound

wound, 235
Electric waves, 426
Electrolysis, discovery of, 218; Faraday's lawB,

289
Electromagnetism, discovery of, 218
Electromagnets, 221, 283
Electromotive force, 256
Electrons, properties of, 435, 444
Electroplating, 290
Electroscope, gold leaf, 194
Electro8tatic8,Dufay's law,194; Coulomb's law 200
Energy, conservation of, 44, 237, 246; measured by

work, 44; potential, 46; kinetic, 46; heat, 162;

magnetic, 237; electric, 270; of voltfuc cell, 284
Engineering units, 50
Equilibrant, 67
Equilibrium, of forces, 64; of paralld forces, 78;

of body free to rotate, 81
Erg, unit of work, 43; symbol, 43

INDEX

449

Ether, transmits waves, 163; propagates light,
426; transmits magnetic and electric action,

near-sighted
417

Evaporation, 146, 161

Expansion, by heat, 143

Eye, image in, 373; far-sighted, 890;

390, how focused, 389; color
Eyepiece, 394

Falling body, 31

Faraday. Michael, discovery of induced currents,
245; ring, 256; electrobm 289

Field, magnetic, 213

Field magnets, motor, 231; dynamo, 253

Fifth, 333

Fizeau, velocity of light, 424

Flotation, 128

Fluids, elasticity of, 118; pressure transmitted by
119; pressure due to weight of, 121

Flute, 336

Fly-wheels, 97; effectiveness of, 103

Focal length, 377; conjugate, 388

Focus, 377; principal, 385; conjugate, 388

Footrpound, 50

Force, relation to mass and acceleration. 37; unit
defined. 38; symbol for, 39; vectors, 63;
centripetal, 105; magnetic lines of, 214; di-
rection of, 220; and displacement in simple
harmonic motion, 319

Force constant, in simple harmonic motion, 321

Force pump, 114

Forced vibrations, 349; of electrons, 441

Foucault, pendulum, 324

Franklin, Benjamin, theory of electricity, 197; and
lightnmg, 204 .

Fringes, interference, 406

Fundamental, of string, 362

Galileo, acceleration of gravity same for all bodies
39; air has weight, 115; thermometer, 138;
telescope, 397; velocity of light, 423

Galvani, 217

Galvanometer, 227; D' Arson val, 228

Gas engine, 182

Gases, no free surface, 124; expand indefinitely.
124; Boyle's law, 131; change of volume at
constant pressure, 139; change of pressure at
constant volume, 140; coefficient of expan-
sion of, 140; diffusion of, 160; pressure of, 161;
kinetic hypothesis. 162; effect of heating, 162;
heated when compressed, 172; cooled when
they expand and do work, 172

Gay Lussac, law of, 140

Gilbert, William, De Magnete, 192; electrifica-
tion, 192; terrestrial magnetiism, 211

Gram calorie, defined, 144; mechanical equiva-
lent of, 171

Gram, unit of inass, 37

Graph, defined, 18

Gravity, acceleration of, same for all bodies, 39;
center of, 81; measured by pendulum, 323

Gravity cell, 288

Gray, Stephen, conduction, 192

Guericke, Otto von, air pump, 125; electric re-
pulsion, 19?

Gyration, radius of, 103

Handd, pure intonation, 347

Harmonic motion, 317

Harmonics, of string^, 362

Harmony, 357

Heat, quantity of, unit of, 144; specific, 144;
latent, 152; mechanical equivalent, 171; con-
duction and convection of, 158; radiant, 160
162; absorption of, 164; and light, 167; a
form of kinetic energy, 162; enei^ consumed
m engine, 179; Ion in electrical transmis-
sion, 277

Heating, effect of current, 276; electric. 280; sys-
tems, hot water. 169; hot air, 169; steam, 169

Heat waves, emitted by bodies at all tempera-
tures, 164; more intense and complex at
high temperatures, 167

Helmholtz, Heinrich von, musical scale, 345;
resonator, 363; color vision, 417

Henry, Joseph, invents telegraph, 222

Hero of Alexandria, turbine, 185

Hertz. Heinrich. electric waves. 237, 427

Homogeneous waves, 308

Hooke, Robert, air pump, 125

Horse-power, 51; value in ergs. 51

Horse-power hour, value in ergs, 180

Humidity, 151

Hydraulic machines, 120

Hydrogen, critical temperature, 150; in the vol'
taic cell, 285; charge of an atom of, 439

H\i)othesis, 91

Ice, manufacture of, 174

Illumination, intensity of, 398

Ions, in electrolysis, 286; hypothesis. 2S6; charge

on, 439
Image, by pin-hole camera. 371; of a point source.

385; construction of. 386; size and distance

of, 387; ^^rtuaI. 388
Impressed period, 331
Incandescent lamp, 272
Incidence, angle of. 378
Inclination, ang.e of or slope, 19
Inclined plane, 66

450

PHYSICS

Index of refraction 37a

Indicator dia$;rair, 178; gauge for making, 188

Indicator steam 188

Induced charge, 199

Induced currents by magnet, 245; by current, 248

Induction coil, 255

Induction, dectrostatic charging by, 199

Inertia, 47; moment of, 101; determination of

moment of, 102
Insulators, electric, 192 .
Intensity of sound depends 'on amplitude, 337;

law of inverse squares, of, 399
interferenre frin«e3 404; wave leng.h measured

by. 406

Jack screw, 88

Jar. Leyden, 201

Joule, James Prescott, mechanical equivalent of

heat, 171; heating effects of current. 275;

law of electric heating, 276

Kilogram, international standard. 37
Kilogram-meter, 51; force, 51
Kinetic energy, equation for, 47; heat a form of,162
Kirchhoff. Gustav, absorption in spectrum, 414

Latent heat, of steam, 152; of water, 152

Laws, definition of, 91

Leclanche cell, 288

Lens, crystalline in eye, 373; action of in image

formation, 373; angle; 386; convex, 376;

concave, 395
Lenz, Heinrich, law of, 250
Lever, 75; mechanical advantage, 76; work done

by, 76; principle, 77
Leyden jar, 201
Lifting magnets, 283

Light, a wave motion, 370; velocity of, 423
Light waves, red appear at a temperature of, 520

C, 167; similar to heat waves, 167; wave

length of, 406; origm of, 435
Lightning, 205
Line loss, 271
Lines of force, magnetic, 213; current induced

when number is changed, 247
Liquefaction of gases, 173
Liquids in equilibrium, laws of< 124; expansion

of by heat, 143
Locfomotive engine, 175; operation of, 176; work

done by. steam in, 177; efficiency of, 180
Lodestone. 211

Longitudinal waves, 307; sound waves arc, 329
Loops, in stationary waves, 311

Machines, law of, 86; mechanical advantajjc from,
87; hydraulic, 120; dynamo-electric. 233, 252

Magdeburg hemispheres, 125

Magnetos, 254

Magnets. 216; lifting. 283

Magnetic, curves, 213; field. 214; lines of force,

214; circuit, 216; declination, 215; unit pole,

216; force, law of, 216; field of moving

charges, 236; system, energy of, 237
Magnification, of telescope, 394
Major scale, 344
Mass, 36; unit of, 37; rdation to force and accel,

eration, 37; relation to weight, 40; center of.

81; moment of, 106; in simple harmonic.

motion, 321
Masses, comparison of, by units of the same kind,

35; by forces that give same acceleration,

36; by weights. 40
Maxwell, James Gierke, dectric waves, 237
Mechanical advantage, of inclined plane, 66; of

lever, 76; of composite machine, 85; from

law of machines, 87; of screw, 88
Mechanical equivalent of heat, 171 ■
Mdting. 152

Mercury, thermometer, 142; spectrum of, 409
Mersenne, laws of strings, 334
Meter, international standard, 15
Microphone, 259

Microscope, simple, 391; compound, 397
Minimum deviation, 410
Mirror, plane, 378; image in, 378; concave, 382
Molecules, 162

Moment of force, 75; of mass, 106
Moment of inertia, 101; determination of, 102
Momentum, change of measures force, 38
Morse, Samud F. B.. introduces telegraph, 223;

alphabet, 223
Motion, uniform, 19; uniformly accelerated, 21;

trandatory, 29; rotary, 29, 102; Newton's

laws of, 47; wave, 302; simple harmonic, 317
Motions, composition of, 57; resolution of, 62
Motor dectric. 230; alternating current, 258;

synchronous, induction, 258
Musical, tone, 331; intervals, 333; scale, 342;

tones, complexity of, 360
Musschenbroek, Leyden jar, 201

Natural period, 331

Ne\vton, Sir Isaac, acceleration of gravity, same
■ for all bodies, 40; laws of motion, 47; Prin-

cipia, 47; gravity with pendulum, 323;

color, 403
Nodes, in stationary waves, 311
Noise, 331

Objective, of telescope, 394
Octave, 333

INDEX

451

Oersted, Hans ChriBtian, discovery of dectro-
magnetism, 218

Ohm. Georg Simon, 266; unit of resistance, 266;
law of, 267

Opera glass, 396

Optical axis, 384

Ordinate, 18

Organ pipe. 335

Origin, of coordinates, 18; of waves, 300; of
light waves, 435

Oscillatory dfecharge, 203; starts electric waves,
427

Overtones, 362; beats among, 365

Oxygen, critical temperature, 150; and combus-
tion. 244

Page, Charles G.. inventor of induction coil.

255
Paints. 418

Parallel, forces, 78; circuits. 272; beam, 384
Parallelogram of motions, 59
Pascal, 116; principle of, 119
Pendulum, proof that gravity acceleration same

for all bodies, 40; compensated, 157; law of.

321; uses of. 323; Foucault. 324; equation

for. 324
Period of vibration, 305; in simple harmonic

motion, 320; relation to force constant and

mass. 321; of pendulum, 323; natural and

impressed, 331
Permeability. 214
Perpetual motion impossible, 43
Phase, of waves, 304
Photo-chemical rays, 430
Photometer, 398

Piano, 332; overtones of strings. 363
Pigments, 418
Pm-hole camera, 372, 382
Pitch. 332; international standard, 349
Pisa, leaning tower of, 39
Polarization, electrostatic, 197
Pole, magnetic, 211; of earth. 215
Potential, difference of, 268
Potential energy, 46
Pound-force, pound-weight, 50
Power, rate of doing work. 50; horse. 51; electric,

268; transmission of. 277
Pressure, atmospheric, 118; due to weight of

fluid. 121; and volume m gas, 142; of satu-
rated vapor, 146; electric, 256
Prevost, theory of exchanges. 164
Prism, spectrum formed by, 403; dispersion of,

407
Proof plane. 195
Pullleys, 95

Pumps, lifting, 113; force, 114; air, 125; theory

of, 126
Pythagoras, musical scale. 345

Radian, unit angle defined, 98

Radiation, of heat. 160. 162; and absorption. 166

Radium. 439

Raitoad curves, 107

Rainbow, 422

Rays, characteristic, 385; cathode, 436

Reaction equals action, 48, 105, 246

Receiver, telephone. 259; wireless telegraph,
428

Reflection, laws of. 378; angle of. 378. 376;
diffuse. 379; total. 382

Refraction, 374; index of, 375; angle of, 376

Related tones, 365

Relay telegraph. 224

Repulsion, electric 193

Resistance, electrical unit of, 266; laws of, 267

Resistivity, 267; table of, 299

Resolution, of motions, 62, 67; by microscope, 398

Resonance, 330; of air columns, 335; of atoms, 415

Resonators, air columns, 335; Hclmholtz, 363;
piano strings, 363; atoms, 415

Resultant motion, 58; force. 64

Reversible processes. 244

Rods, vibrating, 334

Rotation, defined, 29; axis of, 29; and transla-
tion, units of, compared, 101

Rowland. Henry A., mechanical equivalent of
heat, 171; magnetic effect of moving charge,
236; heating effect of current. 276

Saturated vapor, pressure and temperature of, 146

Scientific method. 91

Screw. 88

Second, defined. 15

Self-induction, 263

Series circuit, 218, 270

Shape of waves, simple. 302; complex, 309

Shop, motor and belt driven, 279

Shunts. 282

Simple harmonic motion, 307; related to circular

motion, 317; forces and displacements in,

318; period, 321
Sine curve, 307, 319
Sine, of an angle, 319
Size, angular of image, 387; linear of image, 387;

of electrons, 438
Slope, of a graph, 18; measure of, 19; of a curved

graph, 25
Snell, law of refraction, 376
Sodium, spectrum of. 409
Solids, expansion of by heat, 143

452

PHYSICS

Sound, 328; a wave motion, 328; waves longi-
tudinal, 329; velocity in air, 330; in water,
340; intensity, 337; quality, 357

Sounder, telegraph, 224

Sounding boards, 337

Spark, electric is oscillatory, 204

Specific heat, 144

Spectacles, 390

Spectroscope, 409

Spectrum, 407; continuous, 413; bright-line, 409;
absorption, 414; complete, 429; information,
' obtained from, 430; dark-line, 414; analysis,
414

Spherical aberration, 392

Spinthariscope, 440

Stability, 82

Standards, length, 15; time, 15; mass, 37; pitch
349; candle. 398

Standpipe, 123

Stationary waves, 310

Steam, 145; pressure and temperature of, 146;
superheated, 149; latent heat of, 152; engine
174; work done by, 176; turbine, 184

Stops, in lens, 392

Storage battery, 290

Strings, laws of vibrating, 334; vibrating give
complex tones, 361

Subdominant triad, 343

Submarine boats, 129

Sun, atmosphere of. 415

Superheated vapor, 149

Surface of liquid level, 124

Symbols, for distance, 15; for time, 15; for veloc-
ity, 16; for acceleration, 21; for mass, 37;
for force, 38; for density, 41; for work or
energy, 43; for angular velocity, 98; for an-
gular acceleration, 98

Syphon, 135

Syren. 333

Tangent of an an^e, 20

Telegraph, relay, 224; sounder, 224; key, 226;

wireless. 428
Telephone, 259

Telescope, astronomical, 393; Galileo's 396,
Temperature, Centigrade scale of, 139; alBolute

143; critical. 150; depends on kinetic energy

of molecules 162; efficiency of steam engine

depends on, 180
Tempered musical scale, 347
Thalcs, lodestone, 191
Theory, 91
Thermometer. Galileo's, 138; air, 141; mercury,

142
Thermostat. 157

Thomson, J. J., electrons, 439; construction of

atom, 445
Three-wire system, 278
Time, unit of, 15

Tone quality, and wave shape, 357
Tones, musical, 331; complex, 360; related,

365
Tonic triad. 343
Tops. 108
Torricelli. 115
Torsional vibrations, 327
Transformer. 256
Translation, defined, 29; and rotation, units of.

compared, 101
Transmitter, telephone, 250
Transmission, alternating current, 280
Tcansvene waves. 307
Triads, tonic, dominant, subdominant, 343; vi-

bration ratios in, 344
Trombone, 336
Troughs, of waves, 301
Tuning fori:, 335
Turbines, steam, 184
Tyndall, John, absorption by water vapor, 165

Ultra-red, violet, 429

Uniform motion, equation of, 18

Unit, length, 15; time, 15; mass, 37; heat, 144;

electric, charge, 201; magnetic pole, 216
Units, C-G-S system of, 15

Vacuum nature abhors, 114; tube, 436
Vapors, 131; saturated, 146; superheated, 149
Vector, 58
Velocity, linear, defined, 16; unit of. 16; angular,

98; of waves, 305; of sound in air. 330; in

wat6r, 340; of light, 423; of electric waves,

427
Vibrating, strings, laws of, 334; give complex

tones, 361; flame. 358
Vibration number. 305
Vibration, source of waves, 301; time of, 305;

of pendulum, 321; source of sound, 328; of

strings and rods. 334; number, 343; forced,

349; of electrons, 440
Violin, 334

Virtual image by lens, 388
Viision. limit of distinct, 390
Visual angle, 391
Volt, 267

Volta. Alessandro, 217
Voltaic cell. 217, 283; energy of, 284; pohirisa'

tion of. 285; local action in, 287; commercial,

287
Voltmeter, 268

INDEX

453

Water, density of, 41; boiling point of, 148;
evaporation of, 145; critical temperature,
150; latent heat of, 152; and climate, 153;
vapor, saturated, 148

Water vapor, formation of, 145; saturated, 148;
absorption of heat by, 165; effect of in at-
mosphere on climate, 165

Watt, unit of electrical power, 269; meters, 270

V/aves, in water. 163, 300; heat, 163; origin of,
300; characteristics of, 301; transverse, longi-
tudmal, 307; stationary, 310; light. 406;
electric, 426

Wave front, 384; 444

Wave length, 302; and phase, 305; and velocity,
306; and length of strings, 339; of different
colors, 406; of heat and dectric waves,
429

Wave motion, 302; sound is. 328; light is, 370

Wave shape, and Um» quality, 357

Weight, 38; relation to mass, 39

Weight and mass, proportionality proved by fall-
ing bodies, 39; by pendulum, 323

Wheatstone bridge, 297

White light, interference in, 406; nature of, 443

Windlass, 85

Wireless telegraphy. 428

Wiring table, 298

Work, relation to force and distance, 42; unit of,
43; measures energy, 44; done by steam in
an engine, 177; by electric current 268

X-rays, 440

Yerkes telescope, 395

Young. Thomas, color vision, 417

Zinc^ in voltaic cell, 217. 244, 287; spectrum of,
409

. » ^ » • i--

To avoid fine, this book should be returned on
or before the date last stamped below

• o

• r
H to d

♦^ 524565

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